Effects of Ground Motion Spatial Variations and Random Site Conditions … · could not capture the torsional response induced eccentric poundings, therefore might lead to inaccurate

Effects of Ground Motion Spatial Variations and Random Site Conditions on Seismic

Responses of Bridge Structures

by

Kaiming BI BEng, MEng

This thesis is presented for the degree of Doctor of Philosophy

of The University of Western Australia

Structural Engineering School of Civil and Resource Engineering

May 2011

DECLARATION FOR THESIS CONTAINING PUBLISHED WORK AND/OR WORK PREPARED FOR PUBLICATION

This thesis contains published work and/or work prepared for publication, which has been co-authored. The bibliographical details of the work and where it appears in the thesis are outlined below. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially varying ground motions. Structural Engineering and Mechanics 2010; 36(1): 111-127. (Chapter 2) The estimated percentage contribution of the candidate is 50%. Bi K, Hao H, Chouw N. Required separation distance between decks and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake Engineering and Structural Dynamics 2010; 39(3):303-323. (Chapter 3) The estimated percentage contribution of the candidate is 60%. Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding. Earthquake Engineering and Structural Dynamics, published online. (Chapter 4) The estimated percentage contribution of the candidate is 60%. Bi K, Hao H. Modelling and simulation of spatially varying earthquake ground motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics, under review. (Chapter 5) The estimated percentage contribution of the candidate is 70%. Bi K, Hao H. Influence of irregular topography and random soil properties on the coherency loss of spatial seismic ground motions. Earthquake Engineering and Structural Dynamics, published online. (Chapter 6) The estimated percentage contribution of the candidate is 80%. Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions. Earthquake Engineering and Structural Dynamics, under review. (Chapter 7) The estimated percentage contribution of the candidate is 70%.

Kaiming Bi Print Name Signature Date Hong Hao Print Name Signature Date Nawawi Chouw Print Name Signature Date Weixin Ren 02/03/2011 Print Name Signature Date

School of Civil and Resource Engineering Abstract The University of Western Australia

i

Abstract

The research carried out in this thesis concentrates on the modelling of spatial variation of

seismic ground motions, and its effect on bridge structural responses. This effort brings

together various aspects regarding the modelling of seismic ground motion spatial

variations caused by incoherence effect, wave passage effect and local site effect, bridge

structure modelling with soil-structure interaction (SSI) effect, and dynamic response

modelling of pounding between different components of adjacent bridge structures.

Previous studies on structural responses to spatial ground motions usually assumed

homogeneous flat site conditions. It is thus reasonable to assume that the ground motion

power spectral densities at various locations of the site are the same. The only variations

between spatial ground motions are the loss of coherency and time delay. For a structure

located on a canyon site or site of varying conditions, local site effect will amplify and filter

the incoming waves and thus further alter the ground motion spatial variations. In the first

part of this thesis (Chapters 2-4), a stochastic method is adopted and further developed to

study the seismic responses of bridge structures located on a canyon site. In this approach,

the spatially varying ground motions are modelled in two steps. Firstly, the base rock

motions are assumed to have the same intensity and are modelled with a filtered Tajimi-

Kanai power spectral density function and an empirical spatial ground motion coherency

loss function. Then, power spectral density function of ground motion on surface of the

canyon site is derived by considering the site amplification effect based on the one-

dimensional seismic wave propagation theory. The structural responses are formulated in

the frequency domain, and mean peak responses are estimated by the standard random

vibration method. The dynamic, quasi-static and total responses of a frame structure

(Chapter 2) and the minimum separation distances between an abutment and the adjacent

bridge deck and between two adjacent bridge decks required in the modular expansion

joint (MEJ) design to preclude pounding during strong ground motion shaking are studied

(Chapter 3). The influence of SSI is also examined (in Chapter 4) by modelling the soil

surrounding the pile foundation as frequency-dependent springs and dashpots in the

horizontal and rotational directions.


ii

A method is proposed to simulate the spatially varying earthquake ground motion time

histories at a canyon site with different soil conditions. This method takes into

consideration the local site effect on ground motion amplification and spatial variations.

The base rock motions are modelled by a filtered Tajimi-Kanai power spectral density

function or a stochastic ground motion attenuation model, and the spatial variations of

seismic waves on the base rock are depicted by a coherency loss function. The power

spectral density functions on the ground surfaces are derived by considering seismic wave

propagations through the local site by assuming the base rock motions consisting of out-

of-plane SH wave and in-plane combined P and SV waves with an incident angle to the

site. The spectral representation method is used to simulate the multi-component spatially

varying earthquake ground motions. It is proven that the simulated spatial ground motion

time histories are compatible with the respective target power spectral density or design

response spectrum at each location individually, and the model coherency loss function

between any two of them. This method can be used to simulate spatial ground motions on

a non-uniform site with explicit consideration of the influences of the specific site

conditions. The simulated time histories can be used as inputs to multiple supports of long-

span structures on non-uniform sites in engineering practice.

Based on the proposed simulation technique, the influences of irregular topography and

random soil properties on coherency loss of spatial seismic ground motions are evaluated.

In the analysis, random soil properties are assumed to follow normal distributions and are

modelled by the one-dimensional random fields in the vertical directions. For each

realization of the random soil properties, spatially varying ground motion time histories are

generated and the mean coherency loss functions are derived. Numerical studies show that

coherency function directly relates to the spectral ratio of transfer functions of the two

local sites, and the influence of randomly varying soil properties at a canyon site on

coherency functions of spatial surface ground motions cannot be neglected.

A detailed 3D finite element analysis of pounding responses between different components

of a two-span simply-supported bridge structure on a canyon site to spatially varying

ground motions are performed. The multi-component spatially varying ground motions are

stochastically simulated as inputs and the numerical studies are carried out by using the

transient dynamic finite element code LS-DYNA. Results indicate that the torsional

response of bridge structures induces eccentric poundings between the adjacent bridge

structures. Traditionally used SDOF model or 2D finite element model of bridge structure


iii

could not capture the torsional response induced eccentric poundings, therefore might lead

to inaccurate pounding response predictions. The detailed 3D finite element model is

needed for a more reliable prediction of earthquake-induced pounding responses between

adjacent structures.

School of Civil and Resource Engineering Table of Contents The University of Western Australia

iv

Table of Contents

ABSTRACT ..............................................................................................................................................I

TABLE OF CONTENTS..................................................................................................................... IV

ACKNOWLEDGEMENTS............................................................................................................... VIII

THESIS ORGANIZATION AND CANDIDATE CONTRIBUTION............................................. IX

PUBLICATIONS ARISING FROM THIS THESIS ........................................................................ XII

LIST OF FIGURES ........................................................................................................................... XIV

LIST OF TABLES .......................................................................................................................... XVIII

CHAPTER 1 ..........................................................................................................................................1-1

INTRODUCTION............................................................................................................................................................. 1-1 1.1 BACKGROUND .................................................................................................................................................... 1-1 1.2 RESEARCH GOALS .............................................................................................................................................. 1-6 1.3 OUTLINE ............................................................................................................................................................. 1-7 1.4 REFERENCES....................................................................................................................................................... 1-7

CHAPTER 2......................................................................................................................................... 2-1

RESPONSE OF A FRAME STRUCTURE ON A CANYON SITE TO SPATIALLY VARYING GROUND MOTIONS....... 2-1 2.1 INTRODUCTION.................................................................................................................................................. 2-2 2.2 BRIDGE AND SPATIAL GROUND MOTION MODEL ......................................................................................... 2-4

2.2.1 BRIDGE MODEL..................................................................................................................................2-4 2.2.2 BASE ROCK MOTION..........................................................................................................................2-5 2.2.3 SITE AMPLIFICATION .........................................................................................................................2-7

2.3 STRUCTURAL RESPONSE EQUATION FORMULATION..................................................................................... 2-8 2.4 MAXIMUM RESPONSE CALCULATION.............................................................................................................2-10 2.5 NUMERICAL RESULTS AND DISCUSSIONS ...................................................................................................... 2-11

2.5.1 EFFECT OF SOIL DEPTH...................................................................................................................2-13 2.5.2 EFFECT OF SOIL PROPERTIES .........................................................................................................2-17 2.5.3 EFFECT OF COHERENCY LOSS ........................................................................................................2-20

2.6 CONCLUSIONS .................................................................................................................................................. 2-22


v

2.7 REFERENCES .....................................................................................................................................................2-22

CHAPTER 3......................................................................................................................................... 3-1

REQUIRED SEPARATION DISTANCE BETWEEN DECKS AND AT ABUTMENTS OF A BRIDGE CROSSING A CANYON

SITE TO AVOID SEISMIC POUNDING ............................................................................................................................3-1 3.1 INTRODUCTION ..................................................................................................................................................3-1 3.2 BRIDGE MODEL ..................................................................................................................................................3-4 3.3 SPATIAL GROUND MOTION MODEL .................................................................................................................3-5

3.3.1 BASE ROCK MOTION ......................................................................................................................... 3-5 3.3.2 SITE AMPLIFICATION......................................................................................................................... 3-7

3.4 STRUCTURAL RESPONSES ...................................................................................................................................3-8 3.5 NUMERICAL RESULTS AND DISCUSSIONS.......................................................................................................3-11

3.5.1 EFFECT OF GROUND MOTION SPATIAL VARIATIONS ................................................................. 3-12 3.5.2 EFFECT OF THE BRIDGE GIRDER FREQUENCY............................................................................ 3-16 3.5.3 EFFECT OF THE LOCAL SOIL SITE CONDITIONS .......................................................................... 3-18

3.6 CONCLUSIONS...................................................................................................................................................3-22 3.7 APPENDIX..........................................................................................................................................................3-23

3.7.1 APPENDIX A: MEAN PEAK RESPONSE CALCULATION................................................................ 3-23 3.7.2 APPENDIX B: CHARACTERISTIC MATRICES .................................................................................. 3-24

3.8 REFERENCES .....................................................................................................................................................3-25

CHAPTER 4......................................................................................................................................... 4-1

INFLUENCE OF GROUND MOTION SPATIAL VARIATION, SITE CONDITION AND SSI ON THE REQUIRED

SEPARATION DISTANCES OF BRIDGE STRUCTURES TO AVOID SEISMIC POUNDING .............................................4-1 4.1 INTRODUCTION ..................................................................................................................................................4-2 4.2 BRIDGE-SOIL SYSTEM.........................................................................................................................................4-4 4.3 METHOD OF ANALYSIS ......................................................................................................................................4-7

4.3.1 DYNAMIC SOIL STIFFNESS ................................................................................................................ 4-7 4.3.2 STRUCTURAL RESPONSE FORMULATION ........................................................................................ 4-8

4.4 NUMERICAL EXAMPLE .....................................................................................................................................4-11 4.4.1 INFLUENCE OF SITE EFFECT AND SSI .......................................................................................... 4-12 4.4.2 INFLUENCE OF GROUND MOTION SPATIAL VARIATION AND SSI............................................. 4-17

4.5 CONCLUSIONS...................................................................................................................................................4-19 4.6 APPENDIX..........................................................................................................................................................4-20

4.6.1 APPENDIX A: ELEMENT FOR [ ])( ωiZ AND )]([ ωiZg .............................................................. 4-20

4.6.2 APPENDIX B: PSDS OF THE REQUIRED SEPARATION DISTANCES ............................................ 4-22 4.7 REFERENCES .....................................................................................................................................................4-22

CHAPTER 5......................................................................................................................................... 5-1


vi

MODELLING AND SIMULATION OF SPATIALLY VARYING EARTHQUAKE GROUND MOTIONS AT A CANYON SITE

WITH MULTIPLE SOIL LAYERS ...................................................................................................................................... 5-1 5.1 INTRODUCTION.................................................................................................................................................. 5-2 5.2 WAVE PROPAGATION THEORY AND SITE AMPLIFICATION EFFECT ............................................................ 5-5 5.3 GROUND MOTION SIMULATION....................................................................................................................... 5-8 5.4 NUMERICAL EXAMPLES ................................................................................................................................... 5-11

5.4.1 AMPLIFICATION SPECTRA ...............................................................................................................5-12 5.4.2 EXAMPLE 1-PSD COMPATIBLE GROUND MOTION SIMULATION...............................................5-13 5.4.3 EXAMPLE 2 -RESPONSE SPECTRUM COMPATIBLE GROUND MOTION SIMULATION................5-22

5.5 CONCLUSIONS .................................................................................................................................................. 5-24 5.6 REFERENCES..................................................................................................................................................... 5-25

CHAPTER 6......................................................................................................................................... 6-1

INFLUENCE OF IRREGULAR TOPOGRAPHY AND RANDOM SOIL PROPERTIES ON COHERENCY LOSS OF SPATIAL

SEISMIC GROUND MOTIONS......................................................................................................................................... 6-1 6.1 INTRODUCTION.................................................................................................................................................. 6-2 6.2 THEORETICAL BASIS .......................................................................................................................................... 6-5

6.2.1 ESTIMATION OF COHERENCY FUNCTION.......................................................................................6-5 6.2.2 ONE-DIMENSIONAL WAVE PROPAGATION THEORY.....................................................................6-6 6.2.3 GROUND MOTION GENERATION.....................................................................................................6-7 6.2.4 RANDOM FIELD THEORY ..................................................................................................................6-9 6.2.5 MONTE-CARLO SIMULATION .........................................................................................................6-10

6.3 NUMERICAL EXAMPLE..................................................................................................................................... 6-11 6.3.1 INFLUENCE OF IRREGULAR TOPOGRAPHY...................................................................................6-15 6.3.2 INFLUENCE OF RANDOM SOIL PROPERTIES .................................................................................6-18 6.3.3 INFLUENCE OF RANDOM VARIATION OF EACH SOIL PARAMETER ............................................6-20

6.4 CONCLUSIONS .................................................................................................................................................. 6-23 6.5 REFERENCES..................................................................................................................................................... 6-23

CHAPTER 7......................................................................................................................................... 7-1

3D FEM ANALYSIS OF POUNDING RESPONSE OF BRIDGE STRUCTURES AT A CANYON SITE TO SPATIALLY

VARYING GROUND MOTIONS ...................................................................................................................................... 7-1 7.1 INTRODUCTION.................................................................................................................................................. 7-2 7.2 METHOD VALIDATION ...................................................................................................................................... 7-5 7.3 BRIDGE MODEL .................................................................................................................................................. 7-9 7.4 SPATIALLY VARYING GROUND MOTIONS...................................................................................................... 7-11 7.5 NUMERICAL EXAMPLE..................................................................................................................................... 7-15

7.5.1 LONGITUDINAL RESPONSE.............................................................................................................7-17 7.5.2 TRANSVERSE AND VERTICAL RESPONSES .....................................................................................7-19 7.5.3 TORSIONAL RESPONSE ....................................................................................................................7-22 7.5.4 RESULTANT POUNDING FORCE .....................................................................................................7-24


vii

7.5.5 STRESS DISTRIBUTIONS................................................................................................................... 7-26 7.6 CONCLUSIONS...................................................................................................................................................7-28 7.7 REFERENCES .....................................................................................................................................................7-28

CHAPTER 8......................................................................................................................................... 8-1

CONCLUDING REMARKS ..............................................................................................................................................8-1 8.1 MAIN FINDINGS ..................................................................................................................................................8-1 8.2 RECOMMENDATIONS FOR FUTURE WORK ......................................................................................................8-3

School of Civil and Resource Engineering Acknowledgements The University of Western Australia

viii

Acknowledgements

I would like to express my deep and sincere gratitude to my supervisor, Winthrop

Professor Hong Hao, who supported me persistently during the period of this research.

Prof. Hao was always there to listen and give advice, which enabled my research work to

move forward continuously. Many of the ideas in this thesis would not have taken shape

without his incisive thinking and insightful suggestions. What I learned from him will

benefit me greatly in the rest of my life.

Many thanks go to Associate Professor Nawawi Chouw from the University of Auckland

in New Zealand for his invaluable suggestions, critical and insightful reviews of some of

the papers involved in this thesis. I would also like to thank Professor Weixin Ren, from

Central South University in China, who introduced me to Professor Hao, so that I have the

opportunity to pursue my study in UWA.

I am indebted to the staff and postgraduate students from School of Civil and Resource

Engineering and Centre for Offshore Foundation Systems (COFS) for their friendship and

diverse help during my study in UWA.

I would like to acknowledge the International Postgraduate Research Scholarship (IPRS)

for providing the financial support to me to pursue this study.

At last, I wish to express my sincere thanks to my parents, my brothers and sisters, for their

constant love and inspiration. Without their support, I could not have done it.

School of Civil and Resource Engineering Thesis Organization and Candidate Contribution The University of Western Australia

ix

Thesis Organization and Candidate Contribution

In accordance with the University of Western Australia’s regulations regarding Research

Higher Degrees, this thesis is presented as a series of papers that have been published,

accepted for publication or submitted for publication but not yet accepted. The

contributions of the candidate for the papers comprising Chapters 2~7 are hereby set

forth.

Paper 1

This paper is presented in Chapter 2, first-authored by the candidate, co-authored by

Winthrop Professor Hong Hao and Professor Weixin Ren, and published as

• Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially

varying ground motions. Structural Engineering and Mechanics 2010; 36(1): 111-127.

The candidate developed a program to study the combined ground motion spatial variation

and local site amplification effect on the seismic responses of a frame structure located on

a canyon site. Under the supervision of Winthrop Professor Hong Hao and Professor

Weixin Ren, the candidate overviewed relevant literature, carried out parametrical studies,

interpreted the results and wrote the paper.

Paper 2


Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and published as

• Bi K, Hao H, Chouw N. Required separation distance between decks and at

abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake

Engineering and Structural Dynamics 2010; 39(3):303-323.

The candidate developed a program to study the combined ground motion spatial variation

and local site amplification effect on the required separation distances between abutments


x

and bridge decks and between two adjacent bridge decks of a two-span simply-supported

bridge structure crossing a canyon site to avoid seismic pounding. Under the supervision of

Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, the candidate

overviewed relevant literature, carried out parametrical studies, interpreted the results and

wrote the paper.

Paper 3


Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and has been

published as

• Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site

condition and SSI on the required separation distances of bridge structures to avoid

seismic pounding. Earthquake Engineering and Structural Dynamics, published online.

This paper is an extension of Paper 2 (Chapter 3). The candidate incorporated soil-

structure interaction effect (SSI) into the program developed in Paper 2. Under the

supervision of Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw,

the candidate conducted a series of analysis, highlighted SSI effect and local site conditions

on the required separation distances to avoid seismic pounding of the bridge structure

investigated in Paper 2, and wrote the paper.

Paper 4


Winthrop Professor Hong Hao, and has been submitted as

• Bi K, Hao H. Modelling and simulation of spatially varying earthquake ground

motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics,

under review.

Under the supervision of Winthrop Professor Hong Hao, the candidate incorporated local

site effect of multiple soil layers into the traditional spatially varying seismic ground motion

simulation technique, developed a program to simulate the multi-component spatially

varying seismic motions on the ground surface of a canyon site, and wrote the paper.

Paper 5


Winthrop Professor Hong Hao, and has been published as


xi

• Bi K, Hao H. Influence of irregular topography and random soil properties on the

coherency loss of spatial seismic ground motions. Earthquake Engineering and

Structural Dynamics, published online.

Based on the program developed in Paper 4 (Chapter 5), the candidate studied the

influence of irregular topography and random soil properties on the lagged coherency loss

function of spatial seismic ground motions. Under the supervision of Winthrop Professor

Hong Hao, the candidate overviewed relevant literature, carried out a parametrical study

and wrote the paper.

Paper 6


Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and has been

submitted as

• Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge

structures at a canyon site to spatially varying ground motions. Earthquake

Engineering and Structural Dynamics, under review.

The candidate simulated the multi-component spatially varying ground motions at the

supports of a two-span simply-supported bridge structure located at a canyon site based on

the program developed in Paper 4 (Chapter 5), established the detail 3D finite element

model of the bridge, and investigated the pounding responses of the bridge structure by

using the transient dynamic finite element code LS-DYNA. Under the supervision of

Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, the candidate

overviewed the relevant literature, carried out a parametrical study, highlighted the effect of

torsional response induced eccentric poundings, and wrote the paper.

I certify that, except where specific reference is made in the text to the work of others, the

contents of this thesis are original and have not been submitted to any other university.

Kaiming Bi

May 2011.

Signature:

School of Civil and Resource Engineering Publications Arising From This Thesis The University of Western Australia

xii

Publications Arising From This Thesis

Journal papers

1. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially


2. Bi K, Hao H, Chouw N. Required separation distance between decks and at



3. Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site


seismic pounding. Earthquake Engineering and Structural Dynamics, published online.

4. Bi K, Hao H. Modelling and simulation of spatially varying earthquake ground

motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics,

under review.

5. Bi K, Hao H. Influence of irregular topography and random soil properties on the


Structural Dynamics, published online.

6. Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge

structures at a canyon site to spatially varying ground motions. Earthquake

Engineering and Structural Dynamics, under review.

7. Liang JZ, Hao H, Wang Y, Bi K. Design earthquake ground motion prediction for

Perth metropolitan area with microtremor measurements for site characterization.

Journal of Earthquake Engineering 2009; 13(7): 997-1028.

8. Bai F, Hao H, Bi K, Li H. Seismic response analysis of transmission tower-line

system on heterogeneous sites to multi-component spatial ground motions.

Advances in Structural Engineering, in print.

School of Civil and Resource Engineering Publications Arising From This Thesis The University of Western Australia

xiii

Conferences papers

1. Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to

avoid seismic pounding between adjacent bridge decks. The 14th World Conference on

Earthquake Engineering, Beijing, China, 2008; 03-03-0026.

2. Bi K, Hao H. Seismic response analysis of a bridge frame at a canyons site in

Western Australia. The 14th World Conference on Earthquake Engineering, Beijing, China,

2008; 03-03-0027.

3. Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform

intensities and frequency content. Australian Earthquake Engineering Society 2008

Conference, Ballarat, Australia, 2008; Paper No 18.

4. Liang JZ, Hao H, Wang Y, Bi K. Site characterization evaluation in Perth

metropolitan area using microtremor array method. Proceedings of the 10th International

Symposium on Structural Engineering for Young Experts, Changsha, China, 2008; 1906-

1911.

5. Bi K, Hao H, Chouw N. Dynamic SSI effect on the required separation distances

of bridge structures to avoid seismic pounding. Australian Earthquake Engineering

Society 2009 Conference, Newcastle, Australia, 2009; Paper No 16.

6. Bi K, Hao H. Analysis of influence of an irregular site with uncertain soil properties

on spatial seismic ground motion coherency. Australian Earthquake Engineering Society

2009 Conference, Newcastle, Australia, 2009; Paper No 17.

7. Hao H, Bi K, Chouw N. Combined ground motion spatial variation and local site

amplification effect on bridge structure responses. 6th International Conference on Urban

Earthquake Engineering, Tokyo, Japan, 2009; 595-600.

8. Bi K, Hao H. Pounding response of adjacent bridge structures on a canyon site to

spatially varying ground motions. Australian Earthquake Engineering Society 2010

Conference, Perth, Australia, 2010; Paper No 2.

9. Bi K, Hao H, Zhang C. Analysis of coupled axial-torsional pounding response of

adjacent bridge structures. The 11th International Symposium on Structural Engineering,

Guangzhou, China, 2010; 1612-1618.

School of Civil and Resource Engineering List of Figures The University of Western Australia

xiv

List of Figures

Figure 2-1. Schematic view of a bridge frame crossing a canyon site.................................... 2-5

Figure 2-2. Filtered ground motion power spectral density function on the base rock...... 2-6

Figure 2-3. Different coherency loss functions.......................................................................2-12

Figure 2-4. Site transfer functions for different soil depths ..................................................2-14

Figure 2-5. Power spectral densities of ground motions on site of different depths ........ 2-14

Figure 2-6. Phase difference caused by seismic wave propagation ...................................... 2-14

Figure 2-7. Normalized dynamic responses for different soil depths..................................2-16

Figure 2-8. Normalized total responses for different soil depths.........................................2-16

Figure 2-9. Dynamic, quasi-static and total responses with mhA 0= and mhB 30= ...... 2-17

Figure 2-10. Soil site transfer function for different soil properties ....................................2-17

Figure 2-11. Power spectral densities of ground motions at sites ........................................ 2-18

Figure 2-12. Phase difference owing to seismic wave propagation...................................... 2-18

Figure 2-13. Normalized dynamic responses for different soil properties.......................... 2-18

Figure 2-14. Normalized total responses for different soil properties................................. 2-19

Figure 2-15. Dynamic, quasi-static and total responses (medium soil at support B).........2-19

Figure 2-16. Normalized dynamic responses for different coherency losses ..................... 2-21

Figure 2-17. Normalized total responses for different coherency losses ............................2-21

Figure 3-1. (a) Schematic view of a bridge crossing a canyon site; (b) structural model..... 3-5

Figure 3-2. Filtered ground motion power spectral density function at base rock.............. 3-8

Figure 3-3. Effect of ground motion spatial variation on the required separation distance

(a) 2Δ , (b) 3Δ , (c) 1Δ .......................................................................................................................3-14

Figure 3-4. Left span frequency response function with respect to the frequency ratios.3-16

Figure 3-5. Effect of vibration frequency on the required separation distance.................. 3-17

Figure 3-6. Effect of soil depth on the required separation distance................................... 3-20

Figure 3-7. Effect of soil properties on the required separation distance ........................... 3-21

Figure 3-8. Ground motion power spectral density functions with.....................................3-22


xv

Figure 4-1. (a) Schematic view of a girder bridge crossing a canyon site...............................4-6

Figure 4-2. Frequency-dependent dynamic stiffness and damping coefficients of the pile

group (a)(b) horizontal direction and (c)(d) rotational direction.......................................... 4-12

Figure 4-3. Influence of site effect and SSI on the required separation distances............. 4-13

Figure 4-4. Site effect on ground motion spatial variations: (a) transfer function,............ 4-15

Figure 4-5. Contribution of SSI to the required separation distances with different soil

conditions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ .............................................................. 4-16


conditions ( 0.21 =f Hz ) (a) 3Δ , (b) 2Δ and (c) 1Δ ............................................................. 4-16

Figure 4-7. Influence of ground motion characteristics and SSI on the required separation

distances ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ................................................................. 4-18

Figure 4-8. Contribution of SSI to the required separation distances with different

coherency loss functions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ..................................... 4-19


coherency loss functions ( 0.21 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ..................................... 4-19

Figure 5-1. A canyon site with multiple soil layers (not to scale) ......................................... 5-12

Figure 5-2. Amplification spectra of site 3, (a) horizontal out-of-plane motion;............... 5-13

Figure 5-3. Generated base rock motions in the horizontal directions............................... 5-16

Figure 5-4. Comparison of power spectral density of the generated base rock acceleration

with model power spectral density ........................................................................................... 5-16

Figure 5-5. Comparison of coherency loss between the generated base rock accelerations

with model coherency loss function......................................................................................... 5-17

Figure 5-6. Generated horizontal out-of-plane motions on ground surface...................... 5-18

Figure 5-7. Comparison of power spectral density of the generated horizontal out-of-plane

acceleration on ground surface with the respective theoretical power spectral density ... 5-19

Figure 5-8. Comparison of the coherency loss functions between base rock motions .... 5-19

Figure 5-9. Generated horizontal in-plane motions on ground surface.............................. 5-20

Figure 5-10. Comparison of power spectral density of the generated horizontal in-plane


Figure 5-11. Generated vertical in-plane motions on ground surface................................. 5-21

Figure 5-12. Comparison of power spectral density of the generated vertical in-plane


Figure 5-13. Generated time histories according to the specified design response spectra ..5-

23


xvi

Figure 5-14. Comparison of the generated acceleration and the target response spectra.5-24

Figure 5-15. Comparison of coherency loss between the generated time histories with the

model coherency loss function ..................................................................................................5-24

Figure 6-1. Schematic view of a layered canyon site................................................................. 6-8

Figure 6-2. A four-layer canyon site with deterministic soil properties (not to scale)....... 6-12

Figure 6-3. Simulated acceleration time histories....................................................................6-13

Figure 6-4. Mean values and standard deviations of the lagged coherency of the horizontal

out-of-plane motion at 0.2, 2.0, 5.0 and 9.0Hz .......................................................................6-14

Figure 6-5. Comparison of the mean lagged coherency on the base rock from 600

simulations with the target model .............................................................................................6-14

Figure 6-6. Comparison of the mean lagged coherency between the surface motions (j, k)

with that of the incident motion on the base rock .................................................................6-17

Figure 6-7. Standard deviations of the lagged coherency on the ground surface .............. 6-17

Figure 6-8. Modulus of the site amplification spectral ratio of two local sites ................... 6-17

Figure 6-9. Amplitudes of the site amplification spectra of two local sites ........................ 6-17

Figure 6-10. Influence of uncertain soil properties on...........................................................6-18

Figure 6-11. Influence of uncertain soil properties on the ....................................................6-19

Figure 6-12. Influence of uncertain soil properties on the ....................................................6-19

Figure 6-13. Influence of each random soil property on the ................................................6-21



Figure 7-1. A typical pounding damage between bridge decks in Chi-Chi earthquake....... 7-4

Figure 7-2. Different models (not to scale): (a) lumped mass model (from [11]); ............... 7-7

Figure 7-3. Structural responses based on different models: (a) relative displacement and7-8

Figure 7-4. (a) Elevation view of the bridge, (b) Cross-section of the bridge girder, ........ 7-10

Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings

........................................................................................................................................................ 7-11

Figure 7-6. First four vibration frequencies and mode shapes of the bridge...................... 7-11

Figure 7-7. Simulated acceleration time histories with soft soil condition and intermediately

correlated coherency loss............................................................................................................ 7-14

Figure 7-8. Simulated displacement time histories with soft soil condition and

intermediately correlated coherency loss..................................................................................7-15

Figure 7-9. Comparison of PSDs between the generated horizontal in-plane motions on

ground surface with the respective theoretical model value .................................................7-15


xvii

Figure 7-10. Multi-component spatially varying inputs at different supports of the bridge .7-

16

Figure 7-11. Influence of pounding effect on the longitudinal displacement response ... 7-18

Figure 7-12. Influence of soil conditions on the longitudinal displacement response...... 7-18

Figure 7-13. Influence of coherency loss on the longitudinal displacement response ..... 7-18

Figure 7-14. Influence of pounding effect on the transverse displacement response ...... 7-20

Figure 7-15. Influence of soil conditions on the transverse displacement response......... 7-20

Figure 7-16. Influence of coherency loss on the transverse displacement response......... 7-20

Figure 7-17. Influence of pounding effect on the vertical displacement response ........... 7-21

Figure 7-18. Influence of soil conditions on the vertical displacement response.............. 7-21

Figure 7-19. Influence of coherency loss on the vertical displacement response.............. 7-22

Figure 7-20. Longitudinal displacements of different nodes to case 2 ground motion.... 7-24

Figure 7-21. Influence of soil conditions on the resultant pounding forces ...................... 7-25

Figure 7-22. Influence of coherency loss on the resultant pounding forces ...................... 7-26

Figure 7-23. Stress distributions in the longitudinal direction at left gap of different cases at

the time when peak resultant pounding force occur (a) Case 1 at t=6.27s, (b) Case 3 at

t=7.63s, (c) Case 4 at t=7.96s and (d) Case 5 at t=8.04s (unit: Pa)...................................... 7-27

School of Civil and Resource Engineering List of Tables The University of Western Australia

xviii

List of Tables

Table 2-1. Parameters for coherency loss functions............................................................... 2-12

Table 2-2. Parameters of base rock and different types of soil............................................. 2-12

Table 3-1. Parameters for coherency loss functions...............................................................3-13

Table 3-2. Parameters for local site conditions ......................................................................3-18

Table 4-1. Parameters for local site conditions. ...................................................................... 4-11

Table 5-1. First two vibration frequencies of the sites........................................................... 5-13

Table 7-1. Parameters for local site conditions. ..................................................................... 7-13

Table 7-2. Different cases studied.............................................................................................7-16

Table 7-3. Mean peak displacements in the longitudinal direction (m). .............................. 7-19

Table 7-4. Mean peak displacements in the transverse direction (m). ................................. 7-21

Table 7-5. Mean peak displacements in the vertical direction (m). ...................................... 7-22

Table 7-6. Mean peak rotational angle (degree). ..................................................................... 7-24

School of Civil and Resource Engineering Chapter 1 The University of Western Australia

1-1

Chapter 1 Introduction

1.1 Background

The term “spatial variation of seismic ground motions” denotes the differences in the

amplitude and phase of seismic motions recorded over extended areas. The spatial

variation of seismic ground motions can result from the relative surface fault-motion for

sites located on either side of a causative fault, solid liquefaction, landslides, and from the

general transmission of the waves from the source through the different earth strata to the

ground surface [1]. This thesis concentrates on the latter cause for the spatial variation of

surface ground motions.

The spatial variation of seismic ground motions has an important effect on the response of

large dimensional structures, such as pipelines, dams and bridges. Because these structures

extended over long distances parallel to the ground, their supports undergo different

motions during an earthquake. Since 1960’s, pioneering studies analyzed the influence of

the spatial variation of the motions on the above-ground and buried structures. At that

time, the different motions at the structures’ supports were attributed to the wave passage

effect, i.e., it was considered that the ground motions propagate with a constant velocity on

the ground surface without any change in their shape. The spatial variation of the motions

was then described by the deterministic time delay required for the wave forms to reach the

further-away supports of the structures. In these early studies, it was recognized that wave

passage effect influence the responses of large dimensional structures significantly.

After the installation of the dense seismography arrays in the late 1970’s to early 1980’s, the

modelling of spatial variation of the seismic ground motions and its effect on the responses

of various structural systems attracted extensive research interest. The array, which has

provided an abundance of data for small and large magnitude events that have been


1-2

extensively studied by engineers and seismologists, is the SMART-1 array, located in

Lotung, Taiwan. The spatial variability studies based on these array data provided valuable

information on the physical causes underlying the variations over extended areas and the

means for its modelling. It is generally recognized that four distinct phenomena give rise to

the spatial variability of earthquake-induced ground motions [2]: (1) incoherence effect due

to scattering in the heterogeneous medium of the ground, as well as due to the

superpositioning of waves arriving from an extended source; (2) wave passage effect results

from the different arrival times of waves at separation stations; (3) local site effect owing to

the spatially varying local soil profiles and the manner in which they influence the

amplitude and frequency content of the base rock motion underneath each station as it

propagates upward; and (4) attenuation effect results from gradual decay of wave

amplitudes with distance due to geometric spreading and energy dissipation in the ground

media. For most of the engineering structures, ground motion attenuation over the

distance comparable to the dimension of the structure is usually not significant [2]. This

study thus concentrates on the influence of the first three factors on the ground motion

spatial variation and bridge structural responses.

These dense arrays usually located on the flat-lying alluvial sites, and the recorded ground

motions were usually regarded as homogeneous, stationary and ergodic random field. The

stochastic characteristics of the spatially varying ground motions can be described by the

auto-power spectral density function, cross-power spectral density function and coherency

loss function. The auto-power spectral densities of the motions are estimated from the

analysis of the data recorded at each station and are commonly referred as point estimates

of the motions. Once the power spectra of the motions at the stations of interest have been

evaluated, a parametric form is fitted to the estimates, generally through a regression

scheme. The most commonly used parametric forms of the auto-power spectral density

function are the Tajimi-Kanai power spectrum model [3]. However, this model is

inadequate to describe the ground displacement, as it yields infinite power for the

displacement as the frequency approaches zero. To correct this, Clough-Penzien suggested

introducing a second filter to modify it, which is known as the filtered Tajimi-Kanai Power

spectrum model [4]. Many stochastic ground motion models [5-7] have also been proposed

by considering the rupture mechanism of the fault and the path effect for transmission of

waves through the media from the fault to the ground surface. The joint characteristics of

the time histories at two discrete locations on the ground surface can be depicted by the

cross-power spectral density function and coherency loss function. By processing the

recorded ground motions at these dense arrays, many empirical [8-12] and semi-empirical


1-3

[13-14] spatial ground motion coherency loss function models have been proposed. These

coherency functions usually consist of two parts, the modulus or called lagged coherency,

which measures the similarity of the seismic motions between the two stations, and the

phase, which describes the wave passage effect. It is generally found that the lagged

coherency decreases smoothly as a function of station separation and wave frequency.

These proposed ground motion spatial variation models can be applied directly to the

stochastic analysis of the linear elastic responses of relatively simple structural models.

Previous stochastic studies of ground motion spatial variation effects on the structural

responses include the analysis of a simply-supported beam [15], continuous beams [16, 17],

an arch with multiple horizontal input [18], an arch with multiple simultaneous horizontal

and vertical excitations [19], a symmetric building structure [20], an asymmetric building

structure [21], and a cable-stayed bridge [22]. It should be noted that all these studies were

based on the ground motion spatial variation models by analyzing the data recorded from

the relatively flat-lying sites, the influence of local site effect was not considered. In reality,

seismic waves will be amplified and filtered when propagating through a local soil site. The

amplifications occur at various vibration modes of the site. Therefore, the energy of surface

motions will concentrate at a few frequencies. The power spectral density function of the

surface motion then may have multiple peaks. These phenomena are not considered in

these traditional models. The combined influences of ground motion spatial variation and

local site effect on the structural responses needs to be studied.

Seismic ground motion spatial variations may result in pounding or even collapse of

adjacent bridge decks owing to the large out-of-phase responses. In fact, poundings

between an abutment and bridge deck or between two adjacent bridge decks were observed

in almost all the major earthquakes [23-27]. Many methods were adopted to reduce the

negative effect of pounding. The most direct way to avoid pounding is to provide adequate

separation distance between adjacent structures. For bridge structures with conventional

expansion joints, a complete avoidance of pounding between bridge decks during strong

earthquakes is often impossible, since the separation gap of an expansion joint is usually a

few centimetres to ensure a smooth traffic flow. Recently, a modular expansion joint (MEJ)

system has been developed, and used in some new bridges. The system allows a large

relative movement between the bridge girders without comprising the bridge’s

serviceability and functionality. Using a MEJ, it is possible to make the gap sufficiently large

to cope with the expected closing girder movement, and consequently completely preclude

pounding between adjacent girders. However, up to now studies of the suitability of such a


1-4

system for mitigating adjacent bridge girder pounding responses and its influence on other

bridge response quantities under earthquake loading are limited. Chouw and Hao took two

independent bridge frames as an example, discussed the influences of SSI and non-uniform

ground motions on the separation distance between two adjoined girders connected by a

MEJ [28] and then introduced a new design philosophy for a MEJ [29]. In these studies the

ground motion spatial variations and soil-structure interaction (SSI) were included, the

influence of local site effect, however, was neglected.

In the first part of this thesis (Chapters 2-4), the combined influences of ground motion

spatial variation and local site effect on a frame structure (Chapter 2) and on a two-span

simply-supported bridge (Chapters 3-4) structure are extensively studied based on a

stochastic method. The abutment and the adjacent bridge deck and/or the two adjacent

bridge decks of these structures are connected by a MEJ. The bridge structure is simplified

as a multi-degree-of-freedom (MDOF) system. The structural responses are stochastically

formulated in the frequency domain and the mean peak responses are calculated. In

particular, Chapter 2 investigates the dynamic, quasi-static and total responses of the frame

structure to various cases of spatially varying ground motions. Chapters 3 and 4 study the

required separation distances that MEJs must provide to avoid seismic pounding during

strong earthquakes, with Chapter 3 presenting the influence of ground motion spatial

variation and local site effects and Chapter 4 highlighting the SSI effect.

As mentioned above, the stochastic analysis of the structural responses is usually applied to

relatively simple structural models and for linear response of the structures owing to its

complexity. For complex structural systems and for nonlinear seismic response analysis,

only the deterministic solution can be evaluated with sufficient accuracy. In this case, the

generation of artificial seismic ground motions is required. An extensive list of publications

addressing the topic of simulations of random processes and fields has appeared in the

literature [11, 30-32]. Most these studies [11, 30-31] assumed the power spectral densities

for various locations are the same, the amplification and filtering effect of local site effect

was neglected. The only study considered different power spectral densities of different

locations was reported by Deodatis [32]. This method is based on a spectral representation

algorithm to generate sample functions of a non-stationary, multivariate stochastic process

with evolutionary power spectrum. The considered varying spectral densities are filtered

white noise functions with different central frequency and damping ratio. This method can

only approximately represent local site effect on ground motions, since local soil conditions

will amplify and filter the incoming waves at various vibration modes of the site as


1-5

mentioned above. This phenomenon, however, cannot be considered by this method [32]

since only one peak corresponding to the fundamental vibration mode of the site is

considered. Moreover, trying to establish an analytical expression for a realistic ground

motion evolutionary power spectrum related to the local site conditions is quite difficult

since very limited information is available on the spectral characteristics of propagating

seismic waves [33]. To incorporate local site effect into the simulation technique is a quite

challenging problem in engineering practice. Chapter 5 presents a method to model and

simulate spatially varying earthquake ground motion time histories at sites with non-

uniform conditions. This approach directly relates site amplification effect with local soil

conditions, and can capture the multiple vibration modes of local site, is believed more

realistically simulating the multi-component spatially varying motions on surface of a

canyon site.

Contrasting to the observations on the flat-lying sites, some researchers [34-35] investigated

the lagged coherency loss function between the sites with different conditions, they found

that the lagged coherency does not show a strong dependence on station separation

distance and wave frequency, and the incoherency is generally higher than that on the flat-

lying sites. These observations suggest that the spatial coherency function measured on

flat-lying sedimentary sites may not provide a good description of spatial ground motion

coherencies on sites with irregular topography. However, at the present, only very limited

recorded spatial ground motion data on sites of different conditions are available. They are

not sufficient to determine the general spatial incoherence characteristics of ground

motions and derive empirical relations to model spatial ground motion variations at a site

with varying site conditions. Moreover, all the previous studies on coherency loss functions

assumed the site characteristics are fully deterministic and homogeneous. However, in

reality, there always exist spatial variations of soil properties and uncertainties in defining

the properties of soils. This results from the natural heterogeneity or variability of soils, the

limited availability of information about internal conditions and sometimes the

measurement errors. These uncertainties associated with system parameters are also likely

to have influence on the lagged coherency loss function [36-37]. Theoretical or analytical

analysis in this field is also limited and is in demand. Chapter 6 evaluates the influences of

local site irregular topography and random soil properties on the coherency function

between spatial surface motions based on the approach proposed in Chapter 5.

For bridge structures with conventional expansion joints, a complete avoidance of

pounding between bridge decks during strong earthquakes is often impossible as discussed


1-6

above. Pounding is an extreme complex phenomenon involving damage due to plastic

deformation at contact points, local cracking or crushing, fracturing due to impact, and

friction, etc. To simplify the analysis, many researchers modelled a bridge girder as a

lumped mass [38-43], some other researchers modelled the bridge girders as beam-column

elements [44-45]. Based on theses simplified models, only point to point pounding in 1D,

usually in the axial direction of the structures, can be considered. In reality, pounding could

occur along the entire surfaces of the adjacent structures. Moreover, it was observed from

previous earthquakes that most poundings actually occurred at corners of adjacent bridge

decks. This is because torsional responses of the adjacent decks induced by spatially varying

transverse ground motions at multiple bridge supports resulted in eccentric poundings. To

more realistically model the pounding phenomenon between adjacent bridge structures, a

detailed 3D finite element analysis is necessary. Moreover, ground motion spatial variation,

besides bridge structural vibration properties, is a source of pounding responses in strong

earthquakes. Owing to the difficulty in modelling ground motion spatial variation, many

studies assumed uniform excitations [38, 40-41] or assumed variation was caused by wave

passage effect only [39, 44], only a few studies considered combined wave passage effect

and coherency loss effect in analyzing relative displacement responses of adjacent bridge

structures [42-43, 45]. Study of the combined influences of ground motion spatial variation

and local site effects on earthquake-induced pounding responses of adjacent bridge

structures have not been reported. Chapter 7 investigates the pounding responses between

the abutment and the adjacent bridge deck and between two adjacent bridge decks of a

two-span simply-supported bridge located on a canyon site based on a detailed 3D finite

element model. The influences of local soil conditions and ground motion spatial variations

on the pounding responses are investigated in detail. The influence of torsional response

induced eccentric pounding is highlighted.

1.2 Research goals

This study was undertaken with the aims of:

1. Investigating the combined influences of ground motion spatial variation and local

site effect on the responses of a frame structure.

2. A comprehensive study of ground motion spatial variation, local site effect and SSI

on the required separation distances that MEJs must provide to avoid seismic

pounding.

3. Proposing a method to model and simulate spatially varying earthquake ground

motions on a canyon site with different soil conditions.


1-7

4. Evaluating the influence of irregular topography and random soil properties on

coherency loss function of spatial seismic ground motions.

5. Studying torsional response induced eccentric poundings between adjacent bridge

structures during strong earthquakes.

1.3 Outline

This thesis comprises eight chapters. The seven chapters following this introductory

chapter are arranged as follows:

Chapters 2~4 formulate the structural responses in frequency domain based on the

stochastic method. In particular, Chapter 2 investigates the combined ground motion

spatial variation and local site effect on the response of a frame structure located on a

canyon site. Chapter 3 and 4 study the required separation distances MEJs must provide to

avoid seismic pounding during strong earthquakes, with Chapter 3 presenting the influence

of ground motion spatial variation and local site effects and Chapter 4 highlighting the SSI

effect.

Chapters 5~7 study ground motion spatial variations and structural responses in time

domain. Chapter 5 proposes a method to simulate spatially varying ground motions of a

site with varying soil conditions. Chapter 6 investigates the influence of local site effect and

random soil properties on coherency loss of spatial seismic ground motions. Chapter 7

studies the pounding responses of a two-span simply-supported bridge structure located on

a canyon site based on a detailed 3D FE model.

Finally, Chapter 8 summarizes the main outcomes of this research, along with suggestions

for future studies.

1.4 References

1. Zerva A, Zervas V. Spatial variation of seismic ground motions: an overview.

Applied Mechanics Reviews 2002; 56(3): 271-297.

2. Der Kiureghian A. A coherency model for spatially varying ground motions.

Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.

3. Tajimi H. A statistical method of determining the maximum response of a building

structure during an earthquake. Proc. of 2nd World Conference on Earthquake Engineering,

Tokyo, Japan, 1960; 781-796.


1-8

4. Clough RW, Penzien J. Dynamics of Structures. New York: McGraw Hill; 1993. Joyner

WB, Boore DM. Measurement, characterization and prediction of strong ground

motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in Ground

Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.

5. Joyner WB, Boore DM. Measurement, characterization and prediction of strong

ground motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in

Ground Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.

6. Atkinson GM, Boore DM. Evaluation of models for earthquake source spectra in

Eastern North America. Bulletin of the Seismological Society of America 1998; 88(4): 917-

934.

7. Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use

in Perth Western Australia. Soil Dynamics and Earthquake Engineering 2009; 29(5): 909-

924.

8. Loh CH. Analysis of the spatial variation of seismic waves and ground movement

from SMART-1 data. Earthquake Engineering and Structural Dynamics 1985; 13(5): 561-

581.

9. Harichandran RS, Vanmarcke EH. Stochastic variation of earthquake ground

motion in space and time. Journal of Engineering Mechanics 1986; 112(2): 154-174.

10. Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential

ground movement. Earthquake Engineering and Structural Dynamics 1988; 16(4): 583-

596.

11. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and

simulation based on SMART-1 array data. Nuclear Engineering and Design 1989;

111(3):293-310.

12. Harichandran RS. Estimating the spatial variation of earthquake ground motion

from dense array recordings. Structural Safety 1991; 10: 219-233.

13. Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground

motion. Earthquake Engineering and Structural Dynamics 1986; 14(6): 891-908.

14. Somerville PG, McLaren JP, Saikia CK, Helmberger DV. Site-specific estimation of

spatial incoherence of strong ground motion. Earthquake Engineering and Structural

Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special

Publication No. 20, 1988; 188-202.

15. Harichandran RS, Wang W. Response of simple beam to spatially varying

earthquake excitation. Journal of Engineering Mechanics 1988; 114(9): 1526 - 1541.


1-9

16. Harichandran RS, Wang W. Response of indeterminate two–span beam to spatially

varying seismic excitation. Earthquake Engineering and Structural Dynamics 1990; 19(2):

173-187.

17. Zerva A. Response of multi-span beams to spatially incoherent seismic ground

motion. Earthquake Engineering and Structural Dynamics 1990; 19: 819-832.

18. Hao H. Arch responses to correlated multiple excitations. Earthquake Engineering and

Structural Dynamics 1993; 22(5): 389-404.

19. Hao H. Ground-motion spatial variation effects on circular arch responses. Journal

of Engineering Mechanics 1994; 120(11): 2326-2341.

20. Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings.

Engineering Structures 1996; 18(9): 732-740.

21. Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground

motions. Journal of Structural Engineering 1995; 121(11): 1557-1564.

22. Soyluk K, Dumanoglu AA. Spatial variability effects of ground motions on cable-

stayed bridges. Soil Dynamics and Earthquake Engineering 2004; 24(3):241-250.

23. Hall FJ, editor. Northridge earthquake, January 17, 1994. Earthquake Engineering

Research Institute, Preliminary reconnaissance report, EERI-94-01; 1994.

24. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquake on seismic design

and seismic strengthening of highway bridges. Structural Engineering/Earthquake

Engineering JSCE 1996; 13(2): 211-240.

25. Earthquake Engineering Research Institute. Chi-Chi, Taiwan, Earthquake

Reconnaissance Report. Report No.01-02, EERI, Oakland, California. 1999.

26. Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyakarta earthquake in May 27,

2006. Mid-America Earthquake Centre. Report No. 07-02, 57, 2007.

27. Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 0512 China Wenchuan

earthquake on bridges. The 14th World Conference on Earthquake Engineering. Beijing,

China, 2008; S31-006.

28. Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions

in bridge response II: Effect on response with modular expansion joint. Engineering

Structures 2008; 30(1): 154-162.

29. Chouw N, Hao H. Seismic design of bridge structures with allowance for large

relative girder movements to avoid pounding. New Zealand Society for Earthquake

Engineering Conference. Wairakei, New Zealand 2008; Paper No: 10.

30. Shinozuka M. Monte Carlo solution of structural dynamics. Computers & Structures

1972; 2: 855-874.


1-10

31. Conte JP, Pister KS, Mahin SA. Nonstationary ARMA modelling of seismic ground

motions. Soil Dynamics and Earthquake Engineering 1992; 11: 411-426.

32. Deodatis G. Non-stationary stochastic vector processes: seismic ground motion

applications. Probabilistic Engineering Mechanics 1996; 11(3): 149-167.

33. Shinozuka M, Deodatis G. Stochastic process models for earthquake ground

motion. Probabilistic Engineering Mechanics 1988; 3(3): 114-123.

34. Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site

conditions on the spatial incoherence of ground motions. Structural Safety 1991;

10(1):1-13.

35. Liao S, Zerva A, Stephenson WR. Seismic spatial coherency at a site with irregular

subsurface topography. Proceedings of Sessions of Geo-Denver, Geotechnical Special

Publication No. 170, 2007; 1-10.

36. Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion

coherence and strain estimations. Soil Dynamics and Earthquake Engineering 1997; 16:

445-457.

37. Liao S, Li J. A stochastic approach to site-response component in seismic ground

motion coherency model. Soil Dynamics and Earthquake Engineering 2002; 22: 813-

820.

38. Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete

bridges. Journal of Engineering Mechanics 1998; 124(7):794-802.

39. Jankowski R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated

elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics

1998; 27:487-502.

40. Ruangrassamee A, Kawashima K. Relative displacement response spectra with

pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10): 1511-

1538.

41. DesRoches R, Muthukumar S. Effect of pounding and restrainers on seismic

response of multi-frame bridges. Journal of Structural Engineering (ASCE) 2002; 128(7):

860-869.

42. Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on

pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005;

25(10): 717-728.


in bridge response I: Effect on response with conventional expansion joint.

Engineering Structures 2008; 30(1):141-153.


1-11

44. Jankowski R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges

during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-

212.

45. Chouw N, Hao H, Su H. Multi-sided pounding response of bridge structures with

non-linear bearings to spatially varying ground excitation. Advances in Structural

Engineering 2006; 9(1):55-66.


1-12


2-1

Chapter 2 Response of a Frame Structure on a Canyon Site to Spatially Varying Ground Motions

By: Kaiming Bi, Hong Hao and Weixin Ren

Abstract: This paper studies the effects of spatially varying ground motions on the

responses of a bridge frame located on a canyon site. Compared to the spatial ground

motions on a uniform flat site, which is the usual assumptions in the analysis of spatial

ground motion variation effects on structures, the spatial ground motions at different

locations on surface of a canyon site have different intensities owing to local site

amplifications, besides the loss of coherency and phase difference. In the proposed

approach, the spatial ground motions are modelled in two steps. Firstly, the base rock

motions are assumed to have the same intensity and are modelled with a filtered Tajimi-

Kanai power spectral density function and an empirical spatial ground motion coherency

loss function. Then, power spectral density function of ground motion on surface of the

canyon site is derived by considering the site amplification effect based on the one

dimensional seismic wave propagation theory. Dynamic, quasi-static and total responses of

the model structure to various cases of spatially varying ground motions are estimated. For

comparison, responses to uniform ground motion, to spatial ground motions without

considering local site effects, to spatial ground motions without considering coherency loss

or phase shift are also calculated. Discussions on the ground motion spatial variation and

local soil site amplification effects on structural responses are made. In particular, the

effects of neglecting the site amplifications in the analysis as adopted in most studies of

spatial ground motion effect on structural responses are highlighted.

Keywords: site amplification effect; ground motion spatial variation; dynamic responses;

quasi-static responses; total responses.


2-2

2.1 Introduction

Earthquake ground motions at multiple supports of large dimensional structures inevitably

vary owing to seismic wave propagation effects. Many researchers have investigated seismic

ground motion spatial variations. Most of these studies are based on processing the

recorded ground motions at dense seismographic arrays, such as the SMART-1 array. Many

empirical spatial ground motion coherency loss functions have been derived [1-5]. In all

those studies the site under consideration is assumed to be uniform and homogeneous.

Therefore the ground motion power spectral densities at various locations of the site under

consideration are assumed to be the same. In other words, the only variations in spatial

ground motions are loss of coherency and a phase shift owing to seismic wave propagation.

However, this assumption will lead to inaccurate ground motion representation when a site

has varying conditions such as a canyon site as shown in Figure 2-1. At a canyon site, the

spatial ground motions at base rock can still be assumed to have the same power spectral

density, but on ground surface at points A and B the ground motion power spectral

densities will be very different owing to seismic wave propagation through different wave

paths that cause different site amplifications. Uniform ground motion power spectral

density assumption in such a situation may lead to erroneous estimation of structural

responses.

Some researchers have tried to model the effect of local site conditions on earthquake

ground motion spatial variations. Der Kiureghian et al. [6] proposed a transfer function that

implicitly modelled the site effect on seismic wave propagation. In the model, the ground

motion power spectral density function was represented by a site-dependent transfer

function and a white noise spectrum. Typical site-dependent parameters, i.e., the central

frequency and damping ratio for three generic site conditions, namely, firm, medium and

soft site were proposed. The advantage of this model is that it is straightforward to use.

The drawback is it can only approximately represent the local site effects on ground

motions. For example, it is well known that seismic wave will be amplified and filtered

when propagating through a layered soil site. The amplifications occur at various vibration

modes of the site. Therefore, the energy of surface motions will concentrate at a few

frequencies. The power spectral density function of the surface motion then may have

multiple peaks. This phenomenon, however, cannot be considered in Der Kiureghian’s

model since only one peak corresponding to the fundamental vibration mode of the site

can be involved. In a recent study [7], derivations of earthquake ground motion spatial

variation on a site with uneven surface and different geological properties were presented.


2-3

In the latter study, spatial base rock motion was modelled by a Tajimi-Kanai power spectral

density function [8] together with an empirical coherency loss function [4]. Power spectral

density functions of the surface motions were derived based on the one dimensional

seismic wave propagation theory. Compared to the model by Der Kiureghian et al. [6], the

latter study by Hao and Chouw [7] modelled the base rock motion by the Tajimi-Kanai

power spectral density function instead of a white noise, and the seismic wave propagation

and specific site amplification effects were explicitly represented in terms of the site

conditions such as the soil depth and properties. The multiple vibration modes of local site

can be easily considered. Therefore the latter model gives more realistic prediction of local

site effects on seismic ground motions besides explicitly relating the site conditions to

ground motion model.

Previous studies of ground motion spatial variation effects on structural responses include

stochastic response analysis of a simply supported beam [9], continuous beams [10, 11], an

arch with multiple horizontal input [12], an arch with multiple simultaneous horizontal and

vertical excitations [13], a symmetric building structure [14], an asymmetric building

structure [15], and a cable-stayed bridge [16]. Most of these studies assumed linear elastic

responses. Many researchers have also performed time history analysis of structural

responses to spatially varying ground motions. In these studies, both linear elastic,

nonlinear inelastic responses, pounding responses, soil-structure interaction effects were

considered. The spatial ground motion time histories were obtained either by considering

the wave passage effect only [17], or stochastically simulated to be compatible to a selected

empirical coherency loss function [18-22]. In most of these studies, the site was assumed to

be homogeneous and flat, local site effect was not considered.

Using the model developed by Der Kiureghian et al. [6], Zembaty and Rutenburg [23]

derived the displacement and shear force response spectra with consideration of ground

motion spatial variation and site effects. They concluded that site effects modified the

overall behaviour of the multi-supported structure significantly. Dumanogluid and Soyluk

[24] also used this model and analysed responses of a long span structure to spatially

varying ground motions with site effect. It was concluded that although it was difficult to

draw general conclusions because of the limited analyses performed, it was clear that

ground motion spatial variation and site effects significantly affect the structural responses;

considering different site effects at multiple supports generated larger structural responses;

the more significant was the difference between the site conditions at the multiple

supports, the larger was the structural responses. Another study that used this model to


2-4

consider the site effects and ground motion spatial variation was reported by Ates et al.

[25]. Similar conclusions were drawn, i.e., site effects significantly affect structural

responses. Sextos et al. [20, 21] discussed the importance of considering ground motion

spatial variations, site effect, soil-structure interaction and nonlinear inelastic responses in

bridge response analysis and design. They also outlined the possible numerical approaches

for bridge response analysis.

In the present study, the spatial ground motion model with site effect derived by Hao and

Chouw [7] is used to analyse the responses of a bridge frame on a canyon site. Stochastic

method is used to perform parametric analysis in this study. Dynamic, quasi-static and total

structural responses are calculated. The influences of site conditions and ground motion

spatial variations on structural responses are highlighted. Structural responses to uniform

ground motion, to spatial ground motion without considering coherency loss or phase shift

and to spatial ground motion without considering the site effect are calculated and

compared. Discussions on the ground motion spatial variation and site effect in terms of

the site properties on structural responses are made.

2.2 Bridge and spatial ground motion model

2.2.1 Bridge model

Figure 2-1 illustrates the schematic view of a model bridge frame on a canyon site, in which

A and B are the two supports on ground surface, the corresponding points at base rock

are 'A and 'B . jρ , jv , jξ and jh are the density, shear wave velocity, damping ratio and

depth of the soil under support j, respectively, where j represents A or B. The

corresponding parameters on the base rock are Rρ , Rv and Rξ . The deck of the bridge

frame is idealized as a rigid beam supported by two piers. It should be noted that only one

bridge frame is modelled in the present study, the adjacent bridge structures are neglected.

This simplification implies no pounding between adjacent bridge structures is considered.

This is a rational assumption since with the new development of modular expansion joint

(MEJ), which allows a large joint movement and at the same time without impending the

smoothness of traffic flow, completely precluding seismic pounding between adjacent

bridge structures is possible [26]. This means that each bridge frame can vibrate

independently during an earthquake without pounding between adjacent structures. In the

numerical analysis, without losing generality, the viscous damping ratio of the structure is

assumed to be 5% in the present study.


2-5

Figure 2-1. Schematic view of a bridge frame crossing a canyon site

2.2.2 Base rock motion

Assume the amplitudes of the power spectral densities at different locations on the base

rock are the same and in the form of the filtered Tajimi-Kanai power spectral density

function

Γ+−

+

+−== 222222

2224

2222

4

02

4)(4

)2()()()()(

ωωξωωωωξω

ωξωωωωωωω

ggg

ggg

fffPg SHS (2-1)

in which 2)(ωPH is a high pass filter [27], )(0 ωS is the Tajimi-Kanai power spectral

density function [8], gω and gξ are the central frequency and damping ratio of the Tajimi-

Kanai power spectral density function, Γ is a scaling factor depending on the ground

motion intensity, and ωf and ξf are the central frequency and damping ratio of the high

pass filter. In this study, it is assumed that Hzf ff 25.02/ == πω , 6.0=fξ ,

Hzf gg 0.52/ == πω , 6.0=gξ and 32 /022.0 sm=Γ . These values correspond to a peak

ground acceleration (PGA) 0.5g with duration sT 20= [28]. Figure 2-2 shows the power

spectral density of the base rock motion.

A

B

'B'A

Ah

Bh

Soil AAAA v ξρ ,,

Soil BBBB v ξρ ,,

Base rockRRR v ξρ ,,

d

Ak

Bk


2-6

Figure 2-2. Filtered ground motion power spectral density function on the base rock

Ground motion spatial variation at the base rock is modelled with a coherency loss

function [18]

appapp vdiddvdi

BABA eeeeii /)2/()(/ 2

'''' )()( ωπωωαβωωγωγ −−== (2-2)

in which

sradsradsrad

cbacba

/83.62/83.62/314.0

101.02//2

)(>

≤≤

⎩⎨⎧

++++

=ω

ωπωωπωα (2-3)

where a ,b , c and β are constants, d is the distance between the two supports, appv is the

apparent wave propagation velocity. The cross power spectral density function of the

motion at points 'A and 'B on the base rock is thus

)()()( '''' ωγωω iSiSBAgBA

= (2-4)

It should be noted that the above coherency function was obtained by processing recorded

spatial ground motions on ground surface. Here it is used to model spatial variations of

ground motion at base rock. This is because no information about ground motion spatial

variations at the base rock is available. It is believed that seismic wave propagation through


2-7

a heterogeneous soil site will change ground motion spatial variations. The present

assumption may lead to some inaccurate estimation of coherency loss between spatial base

rock motions. Further research into the influence of local site conditions on spatial ground

motion coherency loss is deemed necessary.

2.2.3 Site amplification

Using seismic wave propagation theory presented by Aki and Richards [29], Safak [30]

derived the transfer function for shear wave propagation in a horizontal layer as

( )( ) BorAjiH

iiiriiir

iUiU

jjjjj

jjjj

j

j ==−−−+

−−−+= )(

)21(2exp)(1)21(exp)1(2

)()(

'

ωξωτξξωτξ

ωω (2-5)

where )( ωiU j and )(' ωiU j is the Fourier transform of the motion )(tu j and )(' tu j

on the

ground surface and at the base rock, respectively. Qj 4/1=ξ is the damping ratio

accounting for energy dissipation owing to seismic wave propagation, and Q is the quality

factor; jjj vh /=τ is the wave propagation time from point 'j to j, and jr is the reflection

coefficient for up-going waves

BorAjvvvv

rjjRR

jjRRj =

+

−=

ρρρρ (2-6)

In engineering application, usually the outcrop motion on the rock surface is available,

instead of the base rock motion. The parameters defined above corresponding to the

Tajimi-Kanai power spectral density function in Equation (2-1) also correspond to the

outcrop motion on hard rock. Therefore, the constant 2 in Equation (2-5), which is a

measure of free surface reflection, in the transfer function is dropped. Then it has

( )( ) BorAj

iiiriiir

iHjjjj

jjjjj =

−−−+

−−−+=

)21(2exp)(1)21(exp)1(

)(ξωτξξωτξ

ω (2-7)

The auto and cross power spectral density function at point j and between points A and B

are

)()()()(

)()()(

''*

2

ωωωω

ωωω

iSiHiHiS

BorAjSiHS

BABAAB

gjj

=

== (2-8)


2-8

in which the superscript ‘*’ represents complex conjugate.

The coherency loss between ground motions at points A and B is

( )[ ]

( )[ ]appBABA

BABA

BA

BABA

BA

ABAB

vdii

iiiHiH

iiHiH

SSiSi

/)()(exp)(

)()()(exp)()(

)()()(

)()()()(

''

''

''

22

*

ωωθωθωγ

ωγωθωθωω

ωγωω

ωωωωγ

+−=

−=== (2-9)

where ( )( ))()(Re

)()(Imtan)()( *

*1

ωωωωωθωθ

iHiHiHiH

BA

BABA

−=− is the phase difference of motions at points A

and B owing to wave propagation at the site. This derivation indicates that the wave

propagation through a homogeneous site has no effect on coherency loss )('' ωγ iBA , but it

changes the phase delay between the spatial ground motion at base rock and on ground

surface, and changes the ground motion intensity. It should be noted that this derivation is

based on assumption that site condition is homogeneous, and ground motion is stationary.

In real case, a soil site will not be homogeneous. The soil properties may vary randomly in

space. Moreover, ground motion is not stationary. All these will cause coherency loss in

spatial ground motions. However, study of the influence of local site conditions on spatial

ground motion coherency loss is beyond the scope of the present paper. It should also be

noted that the transfer function expressed in Equation (2-7) is derived for the case with

only one soil layer. If multiple soil layers are under consideration, it can be

straightforwardly extended based on the seismic wave propagation theory as discussed by

Wolf [31].

2.3 Structural response equation formulation

The purpose of this paper is to investigate the ground motion spatial variation and site

effect on responses of multi-supported structures, the soil-structure interaction effect is

thus ignored. Without losing generality, a 3-DOF mathematical model, with one for the

bridge deck and two for the support movements, is used in the present study. Effectively

such structural model represents only a single dynamic mode of vibration with two

additional kinematic degrees of freedom representing the spatial excitations in the

longitudinal direction. The dynamic equilibrium equation can be written as


2-9

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−−++

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000

00

00000000

00000000

B

A

t

BB

AA

BABA

B

A

t

B

A

t

uuv

kkkk

kkkk

uuvc

uuvm

&

&

&

&&

&&

&&

(2-10)

where m is the lumped mass of the bridge deck, vt is the total displacement response, and uA

and uB are the ground displacement at support A and B respectively, kA and kB are the

stiffness of the two columns. The total response consists of dynamic response and quasi-

static response

qst vvv += (2-11)

The quasi-static response can be derived as

[ ] [ ] )(1BBAA

B

ABA

B

ABA

BA

qs uuuu

uu

kkkk

v ϕϕϕϕ +=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡+

= (2-12)

in which )/( and )/( BABBBAAA kkkkkk +=+= ϕϕ . The dynamic response can be obtained

by solving the dynamic equilibrium equation

)()( BBAAqs

BA uumvmvkkvcvm &&&&&&&&& ϕϕ +−=−=+++ (2-13)

Transfer Equation (2-13) into frequency domain, the dynamic response can be obtained by

)]()()[()()()(2

1)(00

220

ωϕωϕωωωωωωξωω

ω iuiuiHiviHivi

iv BBAAsqs

sqs &&&&&&&& +−=−=

+−−

= (2-14)

in which mkk BA /)(0 +=ω is the circular natural vibration frequency of the structure, 0ξ

is the damping ratio, and )( ωiH s is the transfer function of the structure.

The power spectral density function of dynamic, quasi-static and total response can then be

derived as

[ ]{ }[ ]{ }

[ ]( ){ })(Re2)()()(Re2)()()(

)(Re2)()(1)(

)(Re2)()()()(

222

224

222

ωϕϕωϕωϕωω

ωωω

ωϕϕωϕωϕω

ω

ωϕϕωϕωϕωω

iSSSiHSSS

iSSSS

iSSSiHS

ABBABBAAsvvv

ABBABBAAv

ABBABBAAsv

qst

qs

++−+=

++=

++=(2-15)


2-10

in which ‘Re’ denotes the real part of a complex number. In this study, the uniform ground

motion is assumed to be the same as uA. Under uniform ground motion excitation,

Equation (2-15) reduces to

[ ])()(Re2)()()(

)(1)(

)()()(

2

4

2

ωωω

ωωω

ωω

ω

ωωω

Asvvv

Av

Asv

SiHSSS

SS

SiHS

qsuu

tu

qsu

u

−+=

=

=

(2-16)

2.4 Maximum response calculation

Standard random vibration method [32] is used to calculate the mean peak displacement, it

is briefly described in the following.

For a zero mean stationary process x(t) with known power spectral density function )(ωS ,

its m th order spectral moment is defined as

ωωωλω

dSc mm ∫≈ 0

)( (2-17)

where cω is a high cut-off frequency.

The zero mean cross rate v and shape factor of the power spectral density function δ , can

be obtained by

0

21λλ

π=v (2-18)

20

211λλλδ −= (2-19)

the mean peak response can then be calculated by

σ)ln25772.0ln2(max Tv

Tvxe

e += (2-20)


2-11

where T is the duration of the stationary process, 0λσ = is the standard deviation of the

process, and

69.069.01.01.00

)38.063.1()2,1.2max(

45.0

≥<≤<≤

⎪⎩

⎪⎨

⎧−=

δδδ

δδ

vTvT

TTve

(2-21)

In the present study, the high cut-off frequency is taken as 25Hz since it covers the

predominant vibration modes of most engineering structures and the dominant earthquake

ground motion frequencies.

2.5 Numerical results and discussions

The effects of ground motion spatial variations and site conditions on structural responses

are investigated in detail in the present study. Dynamic, quasi-static and total responses of

the structure in Figure 2-1 under different ground motions and site conditions are

calculated. The phase shift effect of spatial ground motion owing to seismic wave

propagation depends on a dimensionless parameter f0td [1, 9, 11, 12], in which f0 is the

structural vibration frequency, and td is the time lag between ground motions at two points

separated by d. In the previous studies without considering the site effect and with a flat

ground surface assumption, td=d/vapp, in which vapp is the apparent wave propagation

velocity corresponding to the spatial motions at the site. In this study, as discussed above,

vertical wave propagation is assumed in the local site, then the time lag between motions at

points A and B on ground surface can be estimated as td=d/vapp+τB-τA, in which τB and τA,

as defined above, are time required for wave to propagate from B’ to B and A’ to A,

respectively. The spatial ground motion phase shift effect is investigated by varying the

vibration frequency f0 of the structure.

The constants of coherency loss function in Equation (2-2) are obtained by processing

recorded motions during Event 45 at the SMART-1 array [18]. It should be noted that this

coherency loss function represents highly correlated ground motions. For comparison, two

modified coherency loss functions are also used in the study, which represent

intermediately and weakly correlated ground motions, respectively. Figure 2-3 shows

different coherency loss functions corresponding to parameters given in Table 2-1. For

spatial ground motion without coherency loss, ( ) 1'' =ωγ iBA in Equation (2-2).


2-12

Table 2-1. Parameters for coherency loss functions

Coherency loss β a b c

highly(event 45) 410109.1 −× 310583.3 −× 510811.1 −×− 410177.1 −×

intermediately 410697.3 −× 210194.1 −× 510811.1 −×− 410177.1 −×

weakly 310109.1 −× 210583.3 −× 510811.1 −×− 410177.1 −×

Figure 2-3. Different coherency loss functions

The main parameters for base rock and soil conditions can be combined together to form a

single coefficient defined as rock/soil impedance ratio [33]

SS

RRSR v

vIρρ

=/ (2-22)

This impedance coefficient reflects the differences between base rock and soil conditions.

Without losing generality, three types of soils are studied in the paper. The corresponding

parameters for the soil layer and the base rock are given in Table 2-2.

Table 2-2. Parameters of base rock and different types of soil

Type )/( 3mkgρ )/( smv ξ SRI /

Base rock 3000 1500 0.05 /

Firm soil 2000 450 0.05 5

Medium soil 1500 300 0.05 10

Soft soil 1500 100 0.05 30


2-13

2.5.1 Effect of soil depth

The effects of soil depth on structural responses are investigated first. Six different soil

depths are discussed, i.e., the bridge frame locates on a flat site with mhh BA 0== ,

mhh BA 30== , mhh BA 50== or locates on a canyon site with mhmh BA 300 == ,

mhmh BA 500 == and mhmh BA 5030 == . To preclude the influence of other

parameters, the soil under both site A and B are assumed to be firm soil with 5/ =SRI , and

the ground motions are assumed to be intermediately correlated.

As shown in Figure 2-4, different soil depths lead to different transfer functions. The peaks

occur at the corresponding vibration modes of the sites. Take h=50m as a example, the

resonant frequencies of the soil layer are ,...5,3,1,4/ == khkvf sk , where sv and h is the

shear wave velocity and depth of the soil layer respectively, obvious peaks can be obtained

at f=2.25, 6.75 and 11.25Hz with smvs /450= and h=50m. The deeper is the soil, the

more flexible is the site, and the lower is the fundamental vibration frequency. The transfer

function directly alters the ground motion power spectral density function on ground

surface as compared to that at the base rock, as shown in Figure 2-5. Motions on ground

surface have a narrower band, but higher peak, as compared to that at the base rock,

indicating the effect of site filtering and amplification on base rock motion. If the ground

surface is flat, the time lag between motions at A and B are the same as those at the base

rock (τB-τA=0) because soil properties are assumed to be the same at the two wave paths.

In this case, wave propagation through the site will not cause further phase difference.

However, if a canyon site is assumed, the time for wave propagating from base rock to

ground surface is different (τB-τA ≠0), which results in an additional phase difference

between motions at A and B, as compared to those at the base rock, as shown in Figure 2-

6.

Dynamic, quasi-static and total responses with varying structural vibration frequencies are

calculated, and normalized by the corresponding responses to uniform excitation, which is

defined as the motion at Point A, as discussed above. Figures 2-7 and 2-8 show the

normalized dynamic responses and total responses with respect to the dimensionless

parameter, f0td, respectively. This parameter measures the relation between phase shift or

time lag of spatial ground motions at points A and B and the fundamental vibration mode

with frequency f0. When a flat site is considered, f0td=f0d/vapp, and the multiple ground


2-14

excitations and the structural vibration mode are in-phase if L0.2,0.10 =dtf , whereas they

are out-of-phase if L5.1,5.00 =dtf for the special case [28, 34].

Figure 2-4. Site transfer functions for different soil depths

Figure 2-5. Power spectral densities of ground motions on site of different depths

Figure 2-6. Phase difference caused by seismic wave propagation

through sites of different depths


2-15

As shown in Figure 2-7, if the site is flat, non-uniform ground motion always reduces the

dynamic responses as compared to the uniform ground motion. The normalized dynamic

responses reach their minimum value at 5.1,5.00 =dtf and maximum value at 0.2,0.10 =dtf

because of the out-of-phase and in-phase ground motion inputs. This observation is the

same as those reported in many previous studies [28, 34]. If a canyon site is assumed with

Point A on base rock and Point B on soil surface, the maximum responses, however, do

not occur at 0.10 =dtf . This is because of the dominance of site amplification effect on

ground motions and resonant responses. The maximum response occurs when the

structure is resonant with the soil site. For example, when mhA 0= and mhB 30= , the

first peak occurs at f0td=0.625, or f0=3.75Hz because td=d/vapp+τB=d/vapp+hB/vB=0.16667

sec. The second peak can be observed when f0=11.25Hz. As shown in Figures 2-4 and 2-5,

the resonant frequencies of the site with soil depth 30m are ,...5,3,1,75.34/ === kkhkvf sk .

If mhA 0= and mhB 50= , the first peak occurs at f0td =0.475, or f0=2.25Hz because

td=0.2111sec. Again as shown in Figures 2-4 and 2-5, 2.25 Hz is the fundamental vibration

frequency of the soil site with depth 50m. The following peaks can also be observed when

resonance occurs. If both point A and B locate on soil surface with mhA 30= and

mhB 50= , the spatial ground motion wave passage effect dominates the site effect on

dynamic structural responses, i.e., the minimum values occur around 5.1,5.00 =dtf , and the

maximum values around 0.2,0.10 =dtf . This is because, although site A and B have

different fundamental vibration modes and different peak values in their respective power

spectral density function as shown in Figure 2-5, the mean peak responses to ground

motion at site A and B are similar to each other because they depend on the spectral

moments as defined above. Therefore, normalization removes the site amplification effects,

which leaves the wave passage effects to govern the normalized dynamic response in this

case. It can also be noted that the normalized dynamic responses are always smaller than

1.0 when wave passage effect dominates, indicating the spatial ground motion phase shift

always results in a reduction in dynamic structural responses. Similar observation has also

been obtained in previous studies [28, 34]. When the vibration frequency of the structure

coincides with the fundamental frequency of the soil layer, however, the normalized peak

dynamic responses can be larger than 1.0, indicating the significance of site amplifications

on ground motions and hence on structural responses. These observations indicate the

importance of considering both the site and the ground motion spatial variation effects in

structural response analysis.


2-16

Quasi-static responses are independent of the fundamental vibration frequency of the

structure (Equations 2-15 and 2-16). The normalized quasi-static responses are therefore

constant for each case with respect to f0td. The normalized total responses are given in

Figure 2-8. As shown, when the dimensionless parameter f0td is less than 1.5, the

normalized total responses are similar to the normalized dynamic responses, indicating

dynamic response dominates the total response. When f0td increases, however, the

normalized responses approach to a constant, equal to the quasi-static response. Neither

spatial ground motion wave passage effect, nor the site amplification effect is prominent.

This is because increasing f0td implies the structure becomes stiffer, as f0 is increased in this

study. The dynamic response is smaller when structure is stiffer. At large f0td, quasi-static

response dominates the total response, as shown in Figure 2-9. This observation indicates

the importance of quasi-static responses for stiff structures.

Figure 2-7. Normalized dynamic responses for different soil depths

Figure 2-8. Normalized total responses for different soil depths


2-17

Figure 2-9. Dynamic, quasi-static and total responses with mhA 0= and mhB 30=

2.5.2 Effect of soil properties

To study the effect of soil properties on ground motion spatial variation and hence on

structural responses, different soil types shown in Table 2-2 are considered. The soil under

point A is assumed to be firm soil ( 5/ =SRI ) and unchanged in all the cases, while soil

under support B varies from firm soil ( 5/ =SRI ) to soft soil ( 30/ =SRI ). The soil depths

are assumed to be mhA 30= and mhB 50= , and the ground motions are intermediately

correlated. Figure 2-10 shows the transfer function at support B for different cases. Figure

2-11 shows the corresponding power spectral density function of motion on ground

surface at Point B. For comparison purpose, the power spectral density function of motion

at Point A is also shown in these two figures. Figure 2-12 shows the phase differences

between motions at Point A and B. The normalized dynamic responses and total responses

are shown in Figures 2-13 and 2-14, respectively.

Figure 2-10. Soil site transfer function for different soil properties


2-18

Figure 2-11. Power spectral densities of ground motions at sites

with different soil properties

Figure 2-12. Phase difference owing to seismic wave propagation

through sites with different soil properties

Figure 2-13. Normalized dynamic responses for different soil properties


2-19

Figure 2-14. Normalized total responses for different soil properties

Figure 2-15. Dynamic, quasi-static and total responses (medium soil at support B)

Figure 2-10 clearly shows again the site effects. As shown, peak value of the transfer

function increases, while the frequency band becomes narrower with the decrease of the

site stiffness. This directly affects the ground motions on ground surface, resulting in

substantial spatial variations between ground motions at Points A and B. Soft soil

( 30/ =SRI ) and medium soil ( 10/ =SRI ) significantly amplifies the ground motions at its

resonant frequencies, firm soil ( 5/ =SRI ) also amplifies ground motions, but at higher

frequencies and with a less extent. As a result, the ground motion power spectral densities

at ground surface are very different as shown in Figure 2-11. Soil properties also affect the

seismic wave propagation velocity and hence the phase difference between motions at

Point A and B. Figure 2-12 shows the phase differences between motions at A and B

owing to wave propagation from base rock to ground surface. It shows that the phase


2-20

differences vary rapidly with respect to frequency. The softer is the soil, the more drastic

variation is the phase difference because the wave velocity is slower.

Again, as shown in Figure 2-13, when the vibration frequency of the structure is low, site

effect dominates the dynamic responses. When the soil properties of site A and B are

different from each other significantly, i.e., the maximum responses occur when the

structure resonates with the soil site. For example, when site B is the medium soil, the first

peak occurs at f0td =0.3, or f0=1.5Hz because td=0.2 sec. As shown in Figures 2-10 and 2-

11, the fundamental vibration frequency for the medium site is 1.5 Hz. When site B is a

soft soil site, the first peak occurs at f0td =0.267, or f0=0.5Hz because td=0.5333 sec, and

the second peak at f0td =0.8, corresponding to the second mode of the site B. Subsequent

peaks of these two cases are associated with the in-phase excitations and the minimum

values are associated with the out-of-phase effect. This is because when the structure

becomes stiffer, the dynamic response and hence the site resonance effect becomes less

significant as compared with the ground motion spatial variation effect. As also can be seen

in Figure 2-13, soft soil amplification effect results in larger dynamic responses, normalized

dynamic responses are usually larger than 1.0 when the responses are dominated by the site

effect, and the results are always less than 1.0 when spatial ground motion phase shift

effect governs the dynamic responses.

Total responses shown in Figure 2-14 follow the similar pattern as that discussed above,

i.e., the normalized total responses are similar to the normalized dynamic responses when

f0td is less than 1.5. However, if the structure is stiff, the dynamic responses are small and

the total responses are dominated by the quasi-static responses, as shown in Figure 2-15

(medium soil at support B).

2.5.3 Effect of coherency loss

To investigate the influence of ground motion spatial variation, different coherency losses

are considered in the paper as shown in Figure 2-3, i.e., highly, intermediately and weakly

correlated coherency loss functions. Moreover, two special cases, i.e., intermediate

coherency loss without considering phase shift ( 0.1)cos( =dtω ), and no coherency loss

( 1)('' =ωγ iBA

), are also considered. All the results are normalized by the corresponding

uniform excitation. For these cases, the canyon site with mhA 30= and mhB 50= is

considered, and medium soil ( 10/ =SRI ) are assumed at both sites A and B. Figure 2-16


2-21

shows the normalized dynamic responses and Figure 2-17 shows the normalized total

responses.

Figure 2-16. Normalized dynamic responses for different coherency losses

Figure 2-17. Normalized total responses for different coherency losses

As shown in Figure 2-16, site effect governs the dynamic responses when f0td <0.5, i.e.,

peak response occurs at the resonant frequency with f0=0.2 Hz and f0td=0.3. However, if

f0td>0.5, spatial ground motion wave passage effect dominates the dynamic responses, i.e.,

the normalized dynamic responses reach their minimum value at 5.1,5.00 =dtf and their

maximum value at 0.2,0.10 =dtf because of the out-of-phase and in-phase ground motion

excitations. The more correlated are the ground motions, the more pronounced are the in-

phase and out-of-phase effects. This means that the influence of site effect is more

significant when the structure is relatively flexible, while spatial ground motion wave

passage effect dominates the dynamic responses when the structure is stiff. If multiple

ground motion phase shift is not considered, normalized peak response occurs at vibration

modes of the soil site, no in-phase or out-of-phase effects are present. For total responses


2-22

as shown in Figure 2-17, similar observations can be drawn, i.e., dynamic response

dominates total response when f0td<1.0, and quasi-static response is more significant when

the structure is stiff.

2.6 Conclusions

This paper studies the combined effects of ground motion spatial variation and local site

conditions on the responses of a bridge frame located on a canyon site. Dynamic, quasi-

static and total responses of the model structure to various cases of spatially varying ground

motions are investigated. Following conclusions can be drawn:

1. Wave propagation through multiple sites with different site conditions causes

further variations of spatial ground motions. Depending on the soil conditions

along each wave path, spatial ground motions at different locations on surface of a

canyon site have different power spectral densities and more pronounced phase

shift as compared to those on the base rock.

2. Local site conditions significantly affect spatial surface ground motions, and hence

the structural responses. The peak dynamic responses occur when the structure

resonates with the site, and when the spatial ground motion and structural vibration

mode are in-phase. The minimum dynamic responses occur when the spatial

ground motion and structural vibration modes are out-of-phase.

3. Dynamic response governs the total response when the structure is flexible, while

quasi-static response dominates it when the structure is stiff.

4. Different site conditions at two structural supports causes more significant spatial

variations of ground motions, and hence larger structural responses.

5. Spatial ground motion coherency loss has a relatively less significant effect on

structural responses when the structure is flexible and the total response is

governed by the dynamic response. However, coherency loss effect is prominent,

especially when the structure is stiff.

6. Uniform site assumption leads to underestimation of spatial variations of ground

motions on a canyon site, and therefore underestimation of structural responses.

2.7 References

1. Bolt BA, Loh CH, Penzien J, Tsai YB, Yeah YT. Preliminary report on the

SMART-1 strong motion array in Taiwan. Report No. UCB/EERC-82-13, University

of California at Berkeley, Berkeley, CA, 1982.


2-23



3. Loh CH, Yeah YT. Spatial variation and stochastic modelling of seismic differential


596.

4. Hao H, Oliveira CS, Penzien J. Multiple station ground motion processing and

simulation based on SMART-1 data. Nuclear Engineering and Design 1989; 111(3): 293-

310.

5. Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions

for application to soil-structure interaction analyses. Earthquake Spectra 1991; 7: 1-

27.



7. Hao H, Chouw N. Modeling of earthquake ground motion spatial variation on

uneven sites with varying soil conditions. The 9th International Symposium on Structural

Engineering for Young Experts, Fuzhou & Xiamen, 2006.


structure during a earthquake. Proceedings of 2nd World Conference on Earthquake

Engineering, Tokyo, Japan 1960; 781-796.

9. Harichandran RS, Wang W. Response of simple beam to spatially varying

earthquake excitation. Journal of Engineering Mechanics 1988; 114(9): 1526 - 1541.

10. Harichandran RS, Wang W. Response of indeterminate two–span beam to spatially

varying seismic excitation. Earthquake Engineering and Structural Dynamics 1990; 19(2):

173-187.

11. Zerva A. Response of multi-span beams to spatially incoherent seismic ground

motion. Earthquake Engineering and Structural Dynamics 1990; 19: 819-832.

12. Hao H. Arch responses to correlated multiple excitations. Earthquake Engineering and


13. Hao H. Ground-motion spatial variation effects on circular arch responses. Journal

of Engineering Mechanics 1994; 120(11): 2326-2341.

14. Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings.


15. Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground

motions. Journal of Structural Engineering 1995; 121(11): 1557-1564.


2-24

16. Dumanoglu AA, Soyluk K. Response of cable-stayed bridge to spatially varying

seismic excitation. 5th International Conference on Structure Dynamics, Munich, Germany,

2002; 1059-1064.


during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-

212.

18. Hao H. Effects of spatial variation of ground motions on large multiply-supported

structures. Report No. UCB/EERC-89-06, University of California at Berkeley,

Berkeley, 1989.

19. Monti G, Nuti C, Pinto E. Nonlinear response of bridges to spatially varying

ground motion. Journal of Structural Engineering 1996; 122: 1147-1159.

20. Sextos AG, Kappos AJ, Patilakis KD. Inelastic dynamic analysis of RC bridges

accounting for spatial variability of ground motion, site effects and soil-structure

interaction phenomena. Part 1: Methodology and analytical tools. Earthquake

Engineering and Structural Dynamics 2003; 32(4): 607-627.

21. Sextos AG, Kappos AJ, Patilakis KD. Inelastic dynamic analysis of RC bridges


interaction phenomena. Part 2: Parametric study. Earthquake Engineering and




25(10): 717-728.

23. Zembaty Z, Rutenburg A. Spatial response spectra and site amplification effect.


24. Dumanoglu AA, Soyluk K. A stochastic analysis of long span structures subjected

to spatially varying ground motions including the site-response effect. Engineering

Structures 2003; 25(10): 1301-1310.

25. Ates S, Dumanoglu AA, Bayraktar A. Stochastic response of seismically isolated

highway bridges with friction pendulum systems to spatially varying earthquake

ground motions. Engineering Structures 2005; 27(13): 1843-1858.



Structures 2008; 30(1): 154-162.

27. Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during

earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley,

Berkeley, 1969.


2-25

28. Hao H. A parametric study of the required seating length for bridge decks during

earthquake”, Earthquake Engineering and Structural Dynamics 1998; 27(1): 91-103.

29. Aki K, Richards PG. Quantitative seismology theory and methods. WH Freeman

and Company, San Francisco, 1980.

30. Safak E. Discrete-time analysis of seismic site amplification. Journal of Engineering

Mechanics 1995; 121(7): 801-809.

31. Wolf JP. Dynamic soil-structure interaction. New Jersey: Prentice-Hall; 1985.

32. Der Kiureghian A. Structural response to stationary excitation”, Journal of Engineering

Mechanics 1980; 106: 1195-1213.

33. Roesset JM. Soil amplification of earthquakes. Numerical methods in geotechnical

engineering, McGraw-Hill, New York, 1977.

34. Hao H, Zhang S. Spatial ground motion effect on relative displacement of adjacent

building structures”, Earthquake Engineering and Structural Dynamics 1999; 28(4): 333-

349.


3-1

Chapter 3 Required separation distance between decks and at abutments of a bridge crossing a canyon site to avoid seismic pounding

By: Kaiming Bi, Hong Hao and Nawawi Chouw

Abstract: Major earthquakes in the past indicated that pounding between bridge decks may

result in significant structural damage or even girder unseating. With conventional

expansion joints it is impossible to completely avoid seismic pounding between bridge

decks, because the gap size at expansion joints is usually not big enough in order to ensure

smooth traffic flow. With a new development of modular expansion joint (MEJ), which

allows a large joint movement and at the same time without impeding the smoothness of

traffic flow, completely precluding pounding between adjacent bridge decks becomes

possible. This paper investigates the minimum total gap that a MEJ must have to avoid

pounding at the abutments and between bridge decks. The considered spatial ground

excitations are modelled by a filtered Tajimi-Kanai power spectral density function and an

empirical coherency loss function. Site amplification effect is included by a transfer

function derived from the one dimensional wave propagation theory. Stochastic response

equations of the adjacent bridge decks are formulated. The effects of ground motion spatial

variations, dynamic characteristics of the bridge and the depth and stiffness of local soil on

the required separation distance are analysed. Soil-structure interaction effect is not

included in this study. The bridge response behaviour is assumed to be linear elastic.

Keywords: required separation distance; MEJ; spatial variation; site effect; dynamic

characteristic; stochastic method

3.1 Introduction

For large-dimensional structures, such as long-span bridges, earthquake ground motions at

different supports are inevitably not the same owing to seismic wave propagation and local

site conditions. Such ground motion spatial variations may result in pounding or even


3-2

collapse of adjacent bridge decks owing to the out-of-phase responses. Poundings between

an abutment and bridge deck or between two adjacent bridge decks were observed in

almost all the major earthquakes, e.g., the 1989 Loma Prieta earthquake and the 1994

Northridge earthquake [1], the 1995 Hyogo-Ken Nanbu earthquake [2], the 1999 Chi-Chi

Taiwan earthquake [3], the 2006 Yogyakarta earthquake [4], and more recently the 2008

Wenchuan earthquake [5].

More and more earthquake engineers have realized the importance of pounding between

adjacent structures. Many methods were adopted to reduce the negative effect of pounding.

The most direct way to avoid pounding is to provide adequate separation distance between

adjacent structures. Most previous studies on structural pounding have been focused on

adjacent buildings. Jeng et al. [6] estimated the building separation required to avoid

pounding by using spectral difference method. Kasai et al. [7] defined “vibration phase”,

and proposed a simplified rule to predict the inelastic vibration phase. Penzien [8]

proposed a formula for evaluating the required separation distances of two buildings, based

on the procedure of equivalent linearization of non-linear hysteric behaviour. Lin [9]

proposed a theoretical solution based on random vibration method to predict the statistics

of separation distance of adjacent buildings. Hao and Zhang [10] investigated the effect of

the spatially varying ground motions on the relative displacement of adjacent buildings.

Seismic design codes such as UBC [11], Australian Earthquake Loading Codes [12] and

Chinese Seismic Design Code [13] also specify the required separation distances between

buildings.


pounding between bridge decks during strong earthquakes is often impossible. This is

because the separation gap of an expansion joint is usually only a few centimetres to ensure

a smooth traffic flow. Many researchers therefore focused on damaging effects of

pounding and strategies to mitigate pounding between bridge decks. Ruangrassamee and

Kawashima [14] studied the relative displacement spectra of two SDOF systems with

pounding effect. DesRoches and Muthukumars [15] investigated pounding effect on the

global response of a multiple-frame bridge. Owing to the difficulty in modelling the spatial

variation of ground motions, both studies assumed uniform ground motions. Jankowski et

al. [16] and Zhu et al. [17] studied the pounding effect of an elevated bridge caused by wave

passage effect. Hao and Chouw [18] and Zanardo et al. [19] considered the pounding effect

of simply supported segmental bridges, and they confirmed that spatial ground variations

can have strong influences. Jankowski et al. [20] studied several approaches for reducing the


3-3

damaging effects of collisions. To mitigate pounding effect some bridge design codes such

as AASHTO [21], CALTRANS [22] and JRA [23] recommended an adjustment of the

vibration properties of adjacent bridge decks so that they have the same or similar

fundamental periods.

A recent study by Chouw and Hao [24] found that even if the adjacent bridge decks have

exactly the same fundamental period, a few centimetre gap size of a conventional

expansion joint is not sufficient to completely preclude poundings because of ground

motion spatial variations and local site conditions. With the new development of a Modular

Expansion Joint (MEJ), which allows large relative movement in the joint, completely

precluding pounding between bridge decks becomes possible [25]. However, only very

limited information on the required separation distances to avoid seismic pounding

between adjacent bridge structures is available. Hao [26] analysed various parameters that

influence the required seating length to prevent bridge deck unseating; Chouw and Hao

[25] studied the influence of soil-structure interaction (SSI) effect on the required

separation distance of two adjacent bridge frames connected by an MEJ; Bi et al. [27]

studied site effect on the required separation distance between two adjacent bridge decks.

In all these studies, the two adjacent bridge decks were independent and modelled as

uncoupled systems, the bearing that connects bridge pier and deck was not considered, and

the multiple bridge piers were assumed resting on a flat site with the same ground motion

intensity.

In this paper, a more realistic bridge model and site conditions are considered in studying

the required separation distance between two adjacent bridge decks and between the bridge

deck and adjacent abutment. The bridge model is illustrated in Figure 3-1. Bearings that

connect bridge decks to pier or abutments are included in the model. Each bridge deck is

modelled as a rigid beam with lumped mass supported on isolation bearings, and the two

adjacent bridge decks are coupled with each other through the pier which is considered as a

linear elastic reinforced concrete column. The bearings located on the pier and the two

abutments provide with their horizontal flexibility and damping the desired isolation of the

bridge girders. The abutments and pier are supported on ground of different elevations.

Site amplification effect is considered in the study by a transfer function derived from the

one dimensional wave propagation theory. Spatial ground motions are modelled by a

filtered Tajimi-Kanai power spectral density function and an empirical coherency loss

function. Stochastic method is used in the study to calculate the required separation

distances between abutment and deck and between two decks. This paper is a continuation


3-4

of the previous work [26, 27]. The primary differences between the present work and

previous works include: (1) A more realistic bridge model with two adjacent decks coupled

on top of the pier instead of two independent bridge girders, and bearings that connect

bridge decks to the pier and abutments are considered in this study; (2) The required

separation distance between abutments and adjacent bridge deck is also investigated in

addition to that between bridge decks; (3) A canyon site with site amplification effect is

considered in this paper instead of a uniform flat ground surface with the same ground

motion intensity at all the bridge supports; (4) More cases with different bridge girder

frequencies and site conditions are considered in this paper. However, it should be noted

that the effect of soil-structure interaction is not included, and only linear elastic responses

are considered.

3.2 Bridge model

Figure 3-1 (a) illustrates the schematic view of a typical bridge crossing a canyon site. Two

decks with length d1 and d2 are supported by four isolation bearings which are connected

to two abutments and one elastic pier. Points A, B and C are the three bridge support

locations on the ground surface, the corresponding points at base rock are 'A , 'B and 'C .

jh is the depth of the soil layer under the jth support, where j represents A, B or C.

Rρ , Rv and Rξ represent density, shear wave velocity and damping ratio of the base rock,

respectively; the corresponding parameters of soil layer are jρ , jv and jξ .

A MEJ is installed between the bridge decks and at the two abutments. A MEJ consists of

two edge beams and several centre beams, and seals to cover the gaps between the beams

and to ensure the watertightness of the joint. Since the seals move with the gap freely

almost without any resistance, a MEJ allows a large movement gap equivalent to the sum of

a number of small gaps between the beams. Therefore using a MEJ is possible to provide

sufficient closing movement between bridge decks to preclude pounding during strong

ground shakings. More detailed information regarding the MEJ can be found in [25]. The

gap required for a MEJ to avoid pounding between bridge decks and at abutments in terms

of the ground motion properties, site conditions and bridge conditions needs to be

determined.

To simplify the analysis, following hypothesises are made: (1) The bridge decks are

assumed to be rigid with lumped mass m1 and m2; (2) The two bearings located on the

abutment and the pier for each deck have the same dynamic characteristics with stiffness


3-5

kb1 and damping 1bc for the left span, kb2 and 2bc for the right span; (3) The pier is modelled

as an elastic column with a lumped mass at pier top, the corresponding stiffness and

damping are kp and cp, respectively; (4) Spatially varying ground motions are considered at

different supports; (5) Soil-structure interaction is not considered in the present paper.

Based on the above assumptions, a 6 DOF model of the bridge with one DOF for each

rigid deck, one for pier, and three for spatial support movements as shown in Figure 3-1

(b), is used in the present study.

Figure 3-1. (a) Schematic view of a bridge crossing a canyon site; (b) structural model

3.3 Spatial ground motion model

3.3.1 Base rock motion

Assuming ground motion intensities at A’, B’ and C’ on the base rock are the same, and the

coherency loss is measured by an empirical coherency loss function. Its power spectral

density is modelled by a filtered Tajimi-Kanai power spectral density function as

Γ+−

+

+−== 222222

222

2222

4

02

/4)/1(/41

)2()()()()(

ggg

gg

fffPg SHS

ωωξωωωωξ

ωξωωωωωωω (3-1)

)(b

1d 2d

1m 2m

3m1bc 2bc1bc

'A

A

B

'B

C

'C

pp ck

1bk 1bk 2bk 2bk

2bc

1Δ 3Δ

BSoilASoil CSoil

)(a1 2

4 36

5B

A

rockBase

2Δ

C

Ah

BhCh


3-6

in which 2)(ωPH is a high pass filter [28], which is applied to filter out energy at zero and

very low frequencies to correct the singularity in ground velocity and displacement power

spectral density functions. )(0 ωS is the Tajimi-Kanai power spectral density function [29],

gω and gξ are the central frequency and damping ratio of the Tajimi-Kanai power spectral

density function, respectively. ωf and ξf are the central frequency and damping ratio of the

high pass filter, respectively. Without losing generality, in this study, it is assumed that

25.02/ == πω fff Hz, 6.0=fξ , 0.52/ == πωggf Hz and gξ =0.6. Γ is a scaling factor

depending on the ground motion intensity, assuming a ground acceleration of duration

T=20 s and peak value (PGA) 0.5g, 022.0=Γ m2/s3 is estimated in this study according to

the standard random vibration approach given in Appendix A. Figure 3-2 shows the power

spectral density of the base rock acceleration and displacement ( 4/ωgd SS = ).

Ground motions at two distant bridge foundations can vary significantly from each other,

because the propagating seismic waves will not arrive at these locations at the same time,

and the geological medium in the wave path can affect the characteristics of the

propagating waves. In the numerical simulation of spatially varying ground motions usually

empirical coherency loss functions are applied. In the present paper, the coherency loss

function at points j’ and n’ (where j, n represents A, B or C) was derived from the SMART-

1 array data by Hao et al. [30] and is modelled in the following form

appnjnjnjappnj vdiddvdinjnj eeeeii /)2/()(/

''''''

2'''''')()( ωπωωαβωωγωγ −−== (3-2)

in which

srad

sradsradcba

cba/83.62

/83.62/314.0101.0

2//2)(

>≤≤

⎩⎨⎧

++++

=ω

ωπωωπωα (3-3)

where a ,b , c and β are constants, ''njd is the distance between points j’ and n’, 1000=appv

m/s is used in the present paper, which is the apparent wave propagation velocity. It

should be noted that the above empirical coherency loss function was derived from the

recorded strong motions on ground surface at the SMART-1 array. It may not be suitable

to model ground motion spatial variations on the base rock. How the base rock motion

varies spatially, however, is not known. In this study, the soil site is modelled as a


3-7

homogeneous medium. It only affects the ground motion intensity and spatial ground

motion phase delay. The cross power spectral density function of the motion at points j’

and n’ on the base rock is thus

)()()( '''' ωγωω iSiS njgnj = (3-4)

3.3.2 Site amplification

Even if the motion intensities, i.e., the power spectral density functions, at different

locations of the base rock are identical, the surface motions would be different due to the

variation in the filtering and amplification effects of the soil layer at the bridge supports.

The effect of site amplification can be represented by a frequency-dependent transfer

function. In the present study, the transfer function of ground motion due to wave

propagation from base rock j’ to ground surface j is based on the one dimensional wave

propagation assumption, and is given in the following form [31]

)21(2

)21(

)(1)1(

)(jj

jj

j iijj

iijj

S eireir

iH ξωτ

ξωτ

ξ

ξω −−

−−

−+

−+= (3-5)

where jjj vh /=τ is the wave propagation time from point j’ to j, and jr is the reflection

coefficient for up-going waves

jjRR

jjRRj vv

vvr

ρρρρ

+−

= (3-6)

The power spectral density function at point j is thus

)()()(2

ωωω gSj SiHSj

= (3-7)

and the cross power spectral density function between j and n is

)()()()( ''* ωωωω iSiHiHiS njSSjn nj

= (3-8)

where the superscript ‘*’ represents complex conjugate.


3-8

(a)

(b)

Figure 3-2. Filtered ground motion power spectral density function at base rock

(a) acceleration, (b) displacement

3.4 Structural responses

With the hypotheses mentioned above, the dynamic equilibrium equation of the system

shown in Figure 3-1 can be written as

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡00

000

000

gbbTsb

sbss

g

ss

g

ss

yy

KKKK

yyC

yyM

&

&

&&

&& (3-9)

where ][ ssM is the diagonal lumped mass matrix, ][ ssC is the viscous damping matrix and

][ ssK is the stiffness matrix corresponding to the structure degrees of freedom. ][ sbK is the

coupling stiffness matrix between the structure degrees of freedom and the support degrees


3-9

of freedom, ][ bbK is that corresponding to the support movements, [ ] { }321 ,, yyyy T = are the

total displacements vector of the structure and [ ] { }321 ,, gggT

g yyyy = are the ground

displacements vector at the bridge supports, and in which the superscript ‘T’ denotes a

matrix transpose. Corresponding characteristic matrices are given in Appendix B.

The total structural response equation can be derived from Equation (3-9) as

]][[]][[]][[]][[ gsbssssss yKyKyCyM −=++ &&& (3-10)

Equation (3-10) can be decoupled into its modal vibration equation as

][][][2 2

gkss

Tk

sbTk

kkkkkk yM

Kqqqϕϕ

ϕωωξ −=++ &&& (3-11)

where kϕ is the kth vibration mode shape of the structure, kq is the kth modal response,

kω and kξ are the corresponding circular frequency and viscous damping ratio,

respectively.

The kth modal response in the frequency domain can be obtained from Equation (3-11) as

)()()(1

ωψωω iyiHiqr

jgjjkkk ∑

=

= (3-12)

in which r is the total number of supports, and

kkk

k iiH

ωωξωωω

21)( 22 +−

= (3-13)

is the kth mode frequency response function.

kss

Tk

jsb

Tk

jk MK

ϕϕϕψ

][][

−= (3-14)

is the quasi-static participation coefficient for the kth mode corresponding to a movement

at support j, ][ jsbK is a vector in coupled stiffness matrix ][ sbK corresponding to support j.


3-10

The structural response of the ith degree of freedom is

)()(1

tqtyl

kk

ik

i ∑=

= ϕ (3-15)

where l is the number of modes considered in the calculation, and ikϕ is the kth mode

shape value corresponding to the ith degree of freedom.

For the system shown in Figure 3-1, the relative displacement between the adjacent bridge

decks is

)()()()()(1

2

1

1212 tqtqtytyt

l

kkk

l

kkk ∑∑

==

−=−=Δ ϕϕ (3-16)

The power spectral density function of 2Δ can then be derived as:

∑∑ ∑ ∑= = = =

Δ ⎥⎦

⎤⎢⎣

⎡−−=

r

jjn

r

n

l

k

l

snssssjkkkk iSiHiHS

1 1 1 1

*21214 )()()()()(1)(

2ωψωϕϕψωϕϕ

ωω (3-17)

where )( ωiS jn is the cross power spectral density function given in Equation (3-8).

Similarly, the relative displacement between the abutment and the deck is

)()()(

)()()(

32

3

11

1

tytyt

tytyt

g

g

−=Δ

−=Δ (3-18)

The power spectral density functions of 1Δ and 3Δ can be formulated in the following form

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡=

∑∑

∑∑ ∑ ∑

= =

= = = =Δ

r

j

l

kjjkkk

r

jjn

r

n

l

k

l

snsssjkkk

iSiH

iSiHiHS

1 11

14

1 1 1 1

*114

)()(Re2

)()()(1)(1

ωψωϕω

ωψωϕψωϕω

ω (3-19)


3-11

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡=

∑∑

∑∑ ∑ ∑

= =

= = = =Δ

r

j

l

kjjkkk

r

jjn

r

n

l

k

l

snsssjkkk

iSiH

iSiHiHS

1 13

24

1 1 1 1

*224

)()(Re2

)()()(1)(3

ωψωϕω

ωψωϕψωϕω

ω (3-20)

where ‘Re’ denotes the real part of a complex quantity. The mean peak responses can be

obtained based on Equations (3-17), (3-19) and (3-20) by using the standard random

vibration method in Appendix A.

3.5 Numerical results and discussions

Numerical calculations are performed on the relative displacement between the abutment

and the bridge deck, and between the two adjacent bridge decks of the bridge model shown

in Figure 3-1 subjected to spatially varying ground motions at a canyon site. It should be

noted that in the case of strong earthquake non-linear behaviour of piers, bearings and

foundations might occur, which will strongly affect the structural responses. However, the

current work mainly concentrates on the effect of ground motion spatial variation and site

amplification. Therefore, only linear elastic response is considered to avoid further

complicating the discussions.

For two independent structures, the frequency ratio is usually used to measure the

vibration properties of the two adjacent structures, and it is found that the frequency ratio

has a great influence on the required separation distance [10, 24-27]. In the present study,

though the two adjacent bridge decks are coupled with each other on the top of the pier,

the uncoupled frequency ratio 12 / ff of the two spans is still used to approximately quantify

the frequency difference of the two decks. This is because the uncoupled vibration

frequency of each span is easy to be determined and has a straightforward physical

meaning. In this paper, the numerical results are presented with respect to the

dimensionless parameter 12 / ff , where π2//2 111 mkf b= and π2//2 222 mkf b= , are the

uncoupled frequency (in Hz) of the left and right spans, respectively.

To simplify the analysis, the cross sections of the two decks of the bridge model shown in

Figure 3-1 are assumed to be the same, with mass per unit length 4102.1 × kg/m, the lengths

of the left and right spans are assumed to be d1=d2=100 m, so the masses of the two decks

are 621 102.1 ×== mm kg. The lumped mass at the top of the pier is 5

3 102×=m kg. The actual

bearing stiffness of a bridge depends on many factors such as the deck dimension and


3-12

weight, bearing types and dimensions, etc. In practice, most commonly used bearings have

stiffness in the range of 6102× N/m to 7106× N/m. For parametrical study, the bearing

stiffness of the left span is assumed to be 61 106×=bk N/m, which corresponds to the

uncoupled frequency of the left span 5.01 =f Hz. The bearing stiffness of the right span

varies from 5102× N/m to 8106× N/m in the present paper to obtain different frequency

ratios 12 / ff . The stiffness of the pier used in the study is 810=pk N/m. The damping

coefficient of the right span varies with the changing stiffness to maintain the damping

ratio unchanged for the system in the calculation. The damping ratio of 5% is used for

bearings and the pier in the study. It should be noted that the assumption of 5% damping

might underestimate that of the bearings. Since increase damping will reduce the structural

response, this assumption might lead to a conservative estimation of the required gaps to

avoid pounding.

3.5.1 Effect of ground motion spatial variations

To investigate the influence of spatially varying ground motions on the required separation

distance, highly, intermediately and weakly correlated ground motions are considered. The

parameters are given in Table 3-1. For comparison, spatial ground motions with

intermediate coherency loss without considering phase shift ( 0.1)/cos( =appvdω ), spatial

ground motions without considering coherency loss ( 0.1)('' =ωγ iBA

, wave passage effect

only) and uniform ground motion ( 0.1)('' =ωγ iBA) are also considered. To preclude the

effect of site amplification, the analysis in this section assumes that the bridge is located on

the base rock, i.e. 0=== CBA hhh m. Previous papers [24-27] considered the required

separation distance between adjacent bridge decks ( 2Δ in the present paper), no paper

regarding the required separation distance between the abutment and the bridge deck ( 3Δ

and 1Δ ) has been reported. For discussion purpose, the sequences of the required

separation distances discussed in the paper are 2Δ , 3Δ and then 1Δ . The effects of ground

motion spatial variations on the mean minimum required separation distance to avoid

seismic pounding are shown in Figure 3-3. The corresponding standard deviations, which

are not shown here, are rather small as compared to the mean peak responses. Therefore,

only the mean peak responses will be presented and discussed hereafter.


3-13

Table 3-1. Parameters for coherency loss functions

Coherency loss β a b c

Highly 410109.1 −× 310583.3 −× 510811.1 −×− 410177.1 −×

Intermediately 410697.3 −× 210194.1 −×

510811.1 −×− 410177.1 −×

Weakly 310109.1 −× 210583.3 −× 510811.1 −×−

410177.1 −×

As shown in Figure 3-3(a), with an assumption of uniform excitation, the relative

displacement between the two adjacent bridge decks ( 2Δ ) is relatively small when the

fundamental frequencies of the adjacent structures are similar, and is zero when 12 ff = .

This is because the vibration modes of the two adjacent spans are exactly the same when

the frequencies coincide with each other [26]. Therefore, there is no relative displacement

between them. These results correspond well with the recommendations of the current

design regulations to adjusting the frequencies of the adjacent bridge spans to close to each

other in order to preclude pounding. The ground motion spatial variation effect is most

significant when 12 / ff is close to unity, weakly correlated ground motions cause larger

relative displacement than highly correlated ground motions. The ground motion spatial

variation effect is, however, not so pronounced if the vibration frequencies of the two

spans differ significantly. In these situations the out-of-phase vibration of the two spans

owing to their different frequencies contributes most to the relative displacement of

adjacent bridge decks. When 5.1/ 12 >ff , the results are almost constant with the increase

of the frequency ratio. This is because when the structure is relatively stiff as compared to

the ground excitation frequency, the dynamic response of the right span is small. The

displacement response of the right span is caused primarily by the quasi-static response

associated with the non-uniform ground displacement at the multiple bridge supports, and

this quasi-static response is independent of the structural frequency, and is a constant once

the ground displacement is defined. As shown in Figure 3-3(a), there is one obvious peak

occurring at 3.0/ 12 =ff . This is because at this frequency ratio, the first modal vibration

frequency of the coupled system is 0.15 Hz, which coincides with the predominant

frequency of ground displacement as shown in Figure 3-2(b). The above observations also

indicate that adjusting the frequencies of adjacent bridge decks alone is not sufficient to

preclude pounding because ground motion spatial variation also induces relative

displacement of adjacent decks.


3-14

(a)

(b)

(c)

Figure 3-3. Effect of ground motion spatial variation on the required separation distance

(a) 2Δ , (b) 3Δ , (c) 1Δ

Very few researchers studied the required separation distance between bridge deck and

abutment to avoid pounding although pounding damages between bridge deck and


3-15

abutment have been observed in many earthquakes in the past. In this study, the relative

displacement between decks and abutments are calculated. The abutment is assumed to be

rigid and has the same displacement of the respective ground motion. For the separation

distance between the right bridge span and abutment ( 3Δ ) as shown in Figure 3-3(b), one

obvious peak can be observed when the modal frequency of the coupled system coincides

with the predominant frequency of the base rock ground displacement as mentioned

above. The effect of ground motion spatial variation is not prominent. Uniform ground

motion assumption gives a good estimation of the relative displacement. This observation

indicates that the phase shift effect with the assumption of apparent wave velocity

1000=appv m/s, and the influence of coherency loss, are not significant in this considered

bridge example.

As for the relative displacement between the left abutment and the deck, although the

stiffness of the left span remains unchanged, 1Δ in Figure 3-3(c) is not a constant and

varies with the change of f2 because of the coupling through the centre pier. As can be seen

in Figure 3-3(c), when the frequency ratio is slightly smaller than 1, the required separation

distance is small. However, when the frequency ratio is slightly larger than unity, maximum

separation distance is required. This is because, as shown in Figure 3-4, when the

uncoupled vibration frequencies of the left span and right span differ from each other,

changing the vibration frequency of the right span has little influence on that of the left

span. However, the coupled vibration frequency of the left span fluctuates suddenly with

the change of the vibration frequency of the right span when f2/f1 is close to unity. It is

interesting to observe that the spatially varying ground motions have positive effects on 1Δ ,

i.e., weakly correlated ground motions result in a smaller required separation distance and

the largest required separation distance corresponds to the uniform ground excitation case.

It should be noted that, the changing of the stiffness of the right span has no influence on

the required separation distance of the left span if the system is uncoupled through the

pier, and 1Δ will be a constant.


3-16

Figure 3-4. Left span frequency response function with respect to the frequency ratios

3.5.2 Effect of the bridge girder frequency

The bearing stiffness of the left span studied above is 61 106×=bk N/m, which makes the

left deck rather flexible ( 5.01 =f Hz). To cover a larger range of possible cases, two other

bearing stiffness for the left span, i.e., 71 104.2 ×=bk N/m, and 7106.9 × N/m, are also

considered. The corresponding frequencies of the left span are =1f 1.0 Hz and 2.0 Hz,

respectively, which represent intermediate and stiff isolated bridge deck for longitudinal

movements. Figure 3-5 shows the results corresponding to the intermediately correlated

spatial ground motions.

As shown in Figures 3-5(a) and (b), the largest separation distance is required when the

modal vibration frequency of the coupled system coincides with the base rock ground

displacement as previously discussed. The recommendation of the current design

regulations to adjust the adjacent spans to have similar vibration frequencies can be applied

when both of them are relatively flexible (Figure 3-5 (a), 5.01 =f Hz and 1.0 Hz). When

one of the spans or both of them are relatively stiff, this recommendation does not

necessarily produce the minimum separation distance. In fact, the required separation

distance almost reaches a constant when 5.0/ 12 >ff if 0.21 =f Hz as shown in Figure 3-

5(a). It is observed again that having the same vibration frequencies of the adjacent spans

does not completely rule out the relative displacement because of the ground motion

spatial variations. For 1Δ , the coupling effect can still be observed when the uncoupled

vibration frequencies of the two adjacent spans are close to each other, but it is less

pronounced with the increasing of the left span frequency as shown in Figure 3-5(c). As

expected, the higher is the uncoupled frequency of the left span, the smaller separation


3-17

distance is required. These observations indicate that the relative displacement depends not

only on the frequency ratio of the adjacent spans, but also on the absolute uncoupled

frequency of the bridge.

(a)

(b)

(c)

Figure 3-5. Effect of vibration frequency on the required separation distance (a) 2Δ , (b) 3Δ , (c) 1Δ


3-18

3.5.3 Effect of the local soil site conditions

Local soil site conditions have great influences on the structural responses because of the

site filtering and amplification effect on the ground motions. To study the influence of

local site effect, three different types of soils are considered, i.e., firm, medium and soft

soil. Table 3-2 gives the corresponding parameters of site conditions. Figure 3-6 shows the

required separation distances 2Δ , 3Δ , and 1Δ corresponding to the intermediately correlated

ground motions and medium soil with different soil depth. Figure 3-7 shows the results

corresponding to different local soil conditions. In these cases, the ground motion is also

assumed to be intermediately correlated. The canyon site is considered, with mhh CA 50== ,

30=Bh m. The soil under the pier is assumed to be firm soil, while soil under the two

abutments varies from firm soil to soft soil, which is represented by ‘FFF’, ‘MFM’ and

‘SFS’, respectively for the three site conditions considered, in which ‘F’ represents firm, ‘M’

medium and ‘S’ soft soil conditions under each support.

Table 3-2. Parameters for local site conditions

Type Density (kg/m3) Shear wave velocity (m/s) Damping ratio

Base rock 3000 1500 0.05

Firm soil 2000 450 0.05

Medium soil 1500 300 0.05

Soft soil 1500 200 0.05

Based on the discussion above, one obvious peak occurs when the structural frequency

coincides with the central frequency of the ground displacement if the bridge locates on the

base rock. When one or all the bridge supports are located on the soil site, additional peaks

can be observed as shown in Figure 3-6(a), (b) and Figure 3-7(a), (b). One example is

shown in Figure 3-6(a) and (b), when 50== CA hh m ( Bh varies from 0 to 50 m), another

obvious peak occurs at 3/ 12 =ff . This is because when 3/ 12 =ff , the corresponding bearing

stiffness of the right span is 72 104.5 ×=bk N/m. It is found that with this stiffness the

second modal vibration frequency of the coupled system is 1.4 Hz, which coincides with

the predominant frequency of the ground motion on the 50m soil site as can be seen in

Figure 3-8(a), indicating resonance occurs at this frequency ratio. Similar conclusions can

be obtained when the soil depth is 30 m, in this case the site vibration frequency is 2.4 Hz.

Another example is shown in Figure 3-7(a) and (b), when soft soil (SFS) is considered, the

second peak appears when 2/ 12 =ff which corresponds to the second modal frequency of

the coupled system at 0.95 Hz. As can be seen from Figure 3-8(b), 0.95Hz is the


3-19

predominant frequency of the soft site. Similar conclusions can be obtained when firm or

medium site is considered. These observations indicate that larger separation distance is

required when the bridge resonates with local site. Comparing Figure 3-6(a) with Figure 3-

6(b), Figure 3-7(a) with Figure 3-7(b), the results show that the local soil site conditions

have a more significant effect on 3Δ than 2Δ , especially when 2.1/ 12 >ff . This is because

3Δ depends on the absolute response of the structure, while 2Δ depends on the relative

response of the bridge girders. The softer site results in larger absolute structural response

( 3Δ ), which slightly increases the relative displacement ( 2Δ ). As for 1Δ , fluctuations can be

seen when 1f and 2f are close to each other because of the coupling effect as previously

discussed. This example also indicates the importance of local site effect on the required

separation distance. For a bridge structure locates on base rock directly (e.g., [26]), only one

peak can be observed, for a bridge locates on a canyon site, more peaks can be obtained

corresponding to different vibration modes of the local site.

It should be noted that all the above results are based on the assumption of a 5% structural

damping ratio, different damping ratio results in different numerical results. Reference [26]

concluded that the required separation distance decreases with the increasing damping

ratio. The present paper focuses on the total gap that a MEJ must have in order to avoid

pounding, and pounding effect is not considered in the present paper, the effect of

damping ratio will be the same as that in the previous paper. Moreover, the above results

indicate that, as expected, the largest relative displacement is generated when the bridge

structure resonates with ground motions. As dominant ground motion frequency is highly

dependent on the site vibration frequencies, a site investigation to determine the site

vibration frequency is recommended to design the bridge structure to avoid resonance.


3-20

(a)

(b)

(c)

Figure 3-6. Effect of soil depth on the required separation distance

(a) 2Δ , (b) 3Δ , (c) 1Δ


3-21

(a)

(b)

(c)

Figure 3-7. Effect of soil properties on the required separation distance

(a) 2Δ , (b) 3Δ , (c) 1Δ


3-22

(a)

(b)

Figure 3-8. Ground motion power spectral density functions with

(a) different soil depth, (b) different soil properties

3.6 Conclusions

For bridge structures with conventional expansion joints, completely precluding pounding

between bridge decks during strong earthquake excitations is often not possible because

the separation gap in a conventional expansion joint is usually only a few centimetres due

to serviceability consideration for smooth traffic flow. With the new development of the

Modular Expansion Joint, which allows for large relative movements in the joint,

completely precluding pounding between adjacent bridge spans becomes possible. This

paper investigates the minimum separation distance required to avoid seismic pounding of

two adjacent bridge decks coupled on the top of the pier through isolation bearings. The

influence of spatial variation of ground excitations, vibration characteristics of the bridge

structure and local soil conditions on the separation distances between the adjacent bridge

decks, between the abutment and the bridge deck are considered. Following conclusions

can be obtained based on the numerical results:


3-23

1. The required separation distance increases when bridge girders resonate with the

local site, or when modal frequency of the bridge coincides with the central

frequency of ground displacement. Site conditions influence the separation distance

significantly. In general, the deeper and the softer is the local site, the larger is the

required separation distance. Effect of spatially varying ground motions can not be

neglected when the adjacent bridge decks have similar vibration frequencies. Less

correlated ground motions require a larger separation distance to avoid pounding

between bridge decks. The coupling effect is significant on the required separation

distance between the deck and abutment when the uncoupled frequency ratio of

the adjacent spans is close to unity.

2. A consideration of the frequency ratio of the adjacent spans alone is not enough to

determine the required separation distance. The absolute frequency of the bridge

also strongly affects the responses. A flexible bridge requires a larger separation

distance. The recommendation of current design regulations to make adjacent

spans have similar vibration frequency can be applied when both of them are

relatively flexible. When one of the spans or both of them are relatively stiff, this

recommendation does not necessarily give the minimum required separation

distance. This regulation underestimates the smallest separation distance required

between the bridge girder and adjacent abutment.

3. The required MEJ total gap depends on the dynamic properties of the participating

adjacent structures and the dynamic behaviour of the supporting subsoil (not

considered in this work) and the spatially varying ground excitations. Sufficient

total gap of a MEJ should be provided in the bridge design to preclude possible

poundings during strong earthquakes.

3.7 Appendix

3.7.1 Appendix A: Mean peak response calculation

Standard random vibration method is used to calculate the mean peak displacement, it is

briefly described in the following [32].

For a zero mean stationary process x(t) with known power spectral density function )(ωS ,

its m th order spectral moment is defined as

ωωωλω

dSc mm ∫≈ 0

)( (A3-1)


3-24

where cω is a high cut-off frequency.

The zero mean cross rate v and shape factor of the power spectral density function,δ , can

be obtained by

0

21λλ

π=v (A3-2)

20

211λλλδ −= (A3-3)

the mean peak response can then be calculated by

σ)ln25772.0ln2(max Tv

Tvxe

e += (A3-4)

where T is the duration of the stationary process, 0λσ = is the standard deviation of the

process, and

69.069.01.01.00

)38.063.1()2,1.2max(

45.0

≥<≤<≤

⎪⎩

⎪⎨

⎧−=

δδδ

δδ

vTvT

TTve

(A3-5)

In the present study, the high cut-off frequency is taken as Hzc 25=ω since it covers the

predominant vibration modes of most engineering structures and the dominant earthquake

ground motion frequencies.

3.7.2 Appendix B: Characteristic matrices

For the bridge model shown in Figure 3-1, the mass matrix is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

3

2

1

000000

][m

mm

M ss (B3-1)


3-25

where 1m , 2m and 3m is the lumped mass of the two bridge decks and the pier,

respectively.

The stiffness matrices can be formulated as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−−−−

=

pbbbb

bb

bb

ss

kkkkkkkkk

K

2121

22

11

2002

][ (B3-2)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

0000

00][ 2

1

p

b

b

sb

kk

kK (B3-3)

where 1bk and 2bk are the bearing stiffness of the left and right spans, respectively, and pk

the corresponding stiffness of the pier.

3.8 References

1. Yashinsky M, Karshenas MJ. Fundamentals of seismic protection for bridges. Earthquake

Engineering Research Institute, 2003.



Engineering JSCE 1996; 13(2):211-240.




2006. Mid-America Earthquake Centre. Report No. 07-02, 57; 2007.


earthquake on bridges. The 14th world conference on earthquake engineering, Beijing,

China, 2008; S31-006.

6. Jeng V, Kasai K, Maison BF. A spectral difference method to estimate building

separations to avoid pounding. Earthquake Spectra 1992; 8(2):201-223.

7. Kasai K, Jagiasi AR, Jeng V. Inelastic vibration phase theory for seismic pounding

mitigation. Journal of Structural Engineering 1996; 122(10):1136-1146.


3-26

8. Penzien J. Evaluation of building separation distance required to prevent pounding

during strong earthquakes. Earthquake Engineering and Structural Dynamics 1997;

26(8):849-858.

9. Lin JH. Separation distance to avoid seismic pounding of adjacent buildings.

Earthquake Engineering and Structure Dynamics 1997; 26(3):395-403.

10. Hao H, Zhang SR. Spatial ground motion effect on relative displacement of

adjacent building structures. Earthquake Engineering and Structural Dynamics 1999;

28(4):333-349.

11. Uniform Building Code (UBC). International Building Officials. Whittier,

California; 1999.

12. AS 1170.4 SAA Earthquake Loading codes. Stands Association of Australia. 1993.

13. Seismic design code for building and structures-GB11-89. Chinese Academy of Building

Research, Beijing, 1989.


pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10):1151-

1138.


response of multi-frame bridges. Journal of Structural Engineering, ASCE 2002;

128(7):860-869.



1998; 27(5):487-502.

17. Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic

performance of elevated bridges with emphasis on pounding of girders. Earthquake


18. Hao H, Chouw N. Response of a RC bridge in WA to simulated spatially varying

seismic ground motions. Australian Journal of Structural Engineering 2008; 8(1):85-97.

19. Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported

bridges to spatially varying earthquake ground motion. Earthquake Engineering and


20. Jankowski R, Wilde K, Fujino Y. Reduction of pounding in elevated bridges during

earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-212.

21. LRFD bridge design specifications and commentary. American Association of State

Highway and Transportation Officials (AASHTO), 1994.


3-27

22. Caltrans Seismic Design Criteria Version l.2. Department of Transportation.

Sacramento, California, 2001.

23. Design specifications for highway bridges—Part V: Seismic Design. Japan Road

Association (JRA), Tokyo, Japan, 2004.






Structures 2008; 30(1):154-162.


earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1):91-103.

27. Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to

avoid seismic pounding of adjacent bridge decks. The 14th world conference on

earthquake engineering, Beijing, China, 2008; 03-03-0026.


earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley.

1969.


structure during an earthquake. Proceedings of 2nd World Conference on Earthquake

Engineering, Tokyo, 1960; 781-796.



111(3):293-310.

31. Hao H, Chouw N. Modeling of earthquake ground motion spatial variation on

uneven sites with varying soil conditions. The 9th International Symposium on Structural

Engineering for Young Experts, Fuzhou, China, 2006; 79-85.

32. Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering

Mechanics 1980; 106(6): 1195-1213.


4-1

Chapter 4 Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding


Abstract: It is commonly understood that earthquake ground excitations at multiple-

supports of large dimensional structures are not the same. These ground motion spatial

variations may significantly influence the structural responses. Similarly, the interaction

between the foundation and the surrounding soil during earthquake shaking also affects

dynamic response of the structure. Most previous studies of ground motion spatial

variation effects on structural responses neglected soil-structure interaction (SSI) effect.

This paper studies the combined effects of ground motion spatial variation, local site

amplification and SSI on bridge responses, and estimates the required separation distances

that modular expansion joints (MEJs) must provide to avoid seismic pounding. It is an

extension of a previous study [1], in which combined ground motion spatial variation and

local site amplification effects on bridge responses were investigated. The present paper

focuses on the simultaneous effect of SSI and ground motion spatial variation on structural

responses. The soil surrounding the pile foundation is modelled by frequency-dependent

springs and dashpots in the horizontal and rotational directions. The peak structural

responses are estimated by using the standard random vibration method. The minimum

total gap between two adjacent bridge decks or between bridge deck and adjacent abutment

to prevent seismic pounding is estimated. Numerical results show that SSI significantly

affects the structural responses, and cannot be neglected.

Keywords: required separation distance; SSI; site effect; ground motion spatial variation;

MEJ


4-2

4.1 Introduction

Observations from past strong earthquakes revealed that for large-dimensional structures

such as long span bridges, pipelines and communication transmission systems, ground

motion at one foundation may significantly differ from that at another. There are many

reasons that may result in the variability of seismic ground motions, e.g., wave passage

effect results from finite velocity of travelling waves; loss of coherency due to multiple

reflections, refractions and super-positioning of the incident seismic waves; site effect

owing to the differences of local soil conditions; additionally to the above, seismic motion

is further modified by the soil surrounding the foundation, which is known as soil-structure

interaction (SSI) effect. Seismic ground motion variations may result in pounding or even

collapse of adjacent bridge decks owing to the out-of-phase responses. In fact, poundings

between an abutment and bridge deck or between two adjacent bridge decks were observed

in almost all the major earthquakes, e.g. the 1994 Northridge earthquake [2], the 1995

Hyogo-Ken Nanbu earthquake [3], the 1999 Chi-Chi Taiwan earthquake [4], the 2006

Yogyakarta earthquake [5], and more recently the 2008 Wenchuan earthquake [6].

Notwithstanding the extensive research carried out over the last 30 years, very limited

studies involve a comprehensive consideration of the coupling effects of ground motion

spatial variation, site amplification and SSI due to the complexity of these problems. In the

early stage, the sweeping assumptions that the foundations are fixed and the ground

motions at all locations are the same prevail. These assumptions are often used in analysis

of pounding responses between adjacent structures. For example, by assuming uniform

ground motion input, Ruangrassamee and Kawashima [7] calculated the relative

displacement spectra of two SDOF systems with pounding effect; DesRoches and

Muthukumar [8] investigated pounding effect on the global response of a multiple-frame

bridge. After installation of the SMART-1 array in Lotung, Taiwan, more and more

researchers realized the importance of variation of ground motions and some researchers

studied one factor or two factors of ground motion spatial variations on structural

responses. Jankowski et al. [9] and Zhu et al. [10] studied the pounding effect of an

elevated bridge caused by spatial ground motions with only wave passage effect. Taking

combined wave passage effect and coherency loss effect into consideration, Hao [11]

investigated the required seating length to prevent bridge deck unseating; Zanardo et al.

[12] carried out a parametric study of the pounding phenomenon of multi-span simply

supported bridges. To model the combined effects of ground motion spatial variation and

multi-site amplification, Der Kiureghian [13] proposed a transfer function that implicitly


4-3

models the site effect on spatial seismic ground motions. Using this model, Dumanogluid

and Soyluk [14] analysed the stochastic responses of a cable-stayed bridge to spatially

varying ground motions with site effect; Ates et al. [15] investigated the effects of spatially

varying ground motions and site amplifications on the responses of a highway bridge

isolated with friction pendulum systems. In all these studies of ground motion spatial

variation effects on structural responses, SSI effects are however neglected. On the other

hand, Wolf [16] presented a uniform approach in the frequency domain to analyse the free-

field response of local site with multiple soil layers and SSI effect on structural responses;

Spyrakos and Vlassis [17] assessed the effect of SSI on the response of seismically isolated

bridge piers; Makris et al. [18] presented an integrated procedure to analyse the problem of

soil-pile-foundation-superstructure interaction and investigated the effect of SSI on the

Painter Street Bridge in California. These studies concentrate on SSI effect, ground motion

spatial variation are not considered. Chouw and Hao [19, 20] studied the influence of SSI

and non-uniform ground motions on pounding between bridge girders, but neglected the

site amplification effect on spatial ground motions in their study. Shrikhande and Gupta

[21] proposed a stochastic approach for the linear analysis of suspension bridges subjected

to earthquake excitations with consideration of ground motion spatial variation and SSI.

The studies with the broadest scope known to the authors are that by Sextos et al. [22, 23],

who implemented ground motion spatial variation, site effect and SSI into a computer code

ASING, and studied the inelastic dynamic responses of RC bridges in time domain.

To preclude pounding effect, the most straightforward approach is to provide sufficient

separation distances between adjacent structures. For bridge structures with conventional

expansion joints, a complete avoidance of pounding between bridge decks during strong

earthquakes is often impossible. This is because the separation gap of an expansion joint is

usually only a few centimetres to ensure a smooth traffic flow. With the new development

of modular expansion joint (MEJ), which allows large relative movement in the joint,

completely precluding pounding between bridge decks becomes possible [24]. Though the

MEJ systems have already been used in many new bridges, very limited information on the

required separation distance that a MEJ should provide to preclude seismic pounding is

available. Chouw and Hao took two independent bridge frames as an example, discussed

the influences of SSI and non-uniform ground motions on the separation distance between

two adjoined girders connected by a MEJ [24] and then introduced a new design

philosophy for a MEJ [25]; In a recent study [1], the authors combined ground motion

spatial variation effect with site effect, studied the minimum total gap that a MEJ must

provide to avoid seismic pounding at the abutments and between bridge decks. It should


4-4

be noted that these studies either neglected site effect [24, 25] or SSI [1]. To the best

knowledge of the authors, the comprehensive consideration of the coupling of ground

motion spatial variation, site effect and SSI on the required separation distances that MEJs

must provide to preclude seismic pounding has not been reported.

The aim of this paper is to study the combined effects of ground motion spatial variation,

site amplification and SSI on relative responses of adjacent bridge structures. It is an

extension of a previous work [1] in which SSI effect is not considered. The present work

therefore focuses on the SSI effect on bridge structural responses. Random vibration

method is adopted in the study. Spatial ground motions on the base rock are assumed to

have the same intensity, which are modelled by a filtered Tajimi-Kanai power spectral

density function. The wave passage effect and coherency loss effect of the spatial ground

motions on the base rock are modelled by an empirical coherency loss function. Site

amplification effect is included by a transfer function derived from the one dimensional

wave propagation theory. SSI effect is modelled by using the substructure approach. The

soil surrounding the pile foundation is modelled by equivalent frequency-dependent

horizontal and rotational spring-dashpot systems. With linear elastic response assumption,

the bridge responses are formulated and solved in the frequency domain. The power

spectral density functions of the relative displacement responses between adjacent bridge

decks and between bridge deck and abutment are derived and their mean peak responses

are estimated. The minimum total gaps between abutment and bridge deck and two

adjacent bridge decks connected by MEJs to avoid seismic poundings are then determined.

The numerical results obtained in this study can be used as references in designing the total

gap of MEJs.

4.2 Bridge-soil system

Figure 4-1 (a) illustrates the schematic view of a typical girder bridge crossing a canyon site.

The superstructure of the bridge is adopted from [1]. The length and total mass for each

deck are d1=d2=100m and m1=m2=1.2×106kg, respectively. The two bearings for each

deck have the same dynamic properties with an effective stiffness kb1 and an equivalent

viscous damping 1bc for the left span, and kb2 and 2bc for the right span. The concrete pier

with a height of L=20m is modelled as an elastic column with a lumped mass m3=2×105kg

at pier top, the lateral stiffness of the pier is kp=108N/m. To simplify the analysis, a

constant damping ratio of 5% is used for bearings and the pier. Different from Reference

[1], where all the foundations are assumed rigidly fixed to the ground surface, the pier in


4-5

the present study is founded on a rigid cap which is supported by a 2×2 pile group with

the diameter of each pile d=0.6m and axis to axis distance between two adjacent piles

s=3m. The length of the pile is l=12m. To preclude seismic pounding between adjacent

bridge structures, an MEJ is installed at the pier and at the two abutments respectively.

The three bridge support locations on the ground surface are denoted as point 1, 2 and 3 as

shown in Figure 4-1(a), the corresponding points at base rock are 1', 2' and 3'. The soil

depth of the three sites is assumed to be 50, 30 and 50m respectively. This paper focuses

on the relative responses in the horizontal direction, and also owing to the fact that the

vertical stiffness of bridge structure is usually substantially larger than that in the horizontal

direction, the structure is assumed to be rigid in the vertical direction in this analysis.

Moreover, because the abutment of a bridge is usually very stiff as compared to the pile

foundation, SSI between foundation and abutment is less significant as compared to that

between pile and the surrounding soil, and is often neglected in the analysis [26]. The SSI

effects between the abutments and the supporting soils are also neglected in this study.

Therefore only the dynamic interaction effect between the pile foundation and the

surrounding soil is considered. The soil surrounding the pile foundation at site 3 is

modelled as springs and dashpots with the frequency-dependent coefficients hk , hc in the

horizontal direction and rk , rc in the rotational direction. Only viscous damping, which is

developed through the energy emanating from the foundation in the soil medium, is

considered. The corresponding values of these coefficients are related to the pile and soil

conditions, which will be discussed in Section 4.3.1. For simplicity, the rigid cap supporting

the pier is assumed massless.

With the above assumptions, the bridge can be modelled as a five-degree-of-freedom

system as shown in Figure 4-1(b): the dynamic displacements 1u and 2u of the bridge deck

movement relative to the free field motion 1gu and 2gu ; the horizontal displacement 0u of

the pile foundation at site 3 relative to the free field motion 3gu ; the rotational response φ

of the pier at the foundation level and the dynamic response 3u at the pier top.


4-6

2m2gu2u1gu 1u

1m

)(b

1d 2d

1m 2m

3m1bc 2bc1bc

'1

1

3

'3

2

'2

pp ck

1bk 1bk 2bk 2bk

2bc

1Δ

3Site1Site 2Site

)(a

rockBase

3Δ

hc

hk

s d

2Δ

L

3gu 0u φL 3u3m

φ

rcrk

Figure 4-1. (a) Schematic view of a girder bridge crossing a canyon site

and (b) structural model


4-7

4.3 Method of analysis

In the present paper, the structural responses are calculated in the frequency domain. Free-

field ground motions are used as input in the analyses. In general, the ground motions at

the supports are different from the free-field seismic motion. These differences are caused

by the scattered wave fields, which generate between soil and structure interface. However,

for motions that are not rich in high frequencies, the scattered fields are weak. The support

motions therefore can be approximately considered equal to the free-filed motions [18, 27].

Similar to Reference [1], the free-field spatial ground motions in the present study are

derived from the base rock motions together with the transfer function of local site. To

avoid repetition, they are not presented here. The detailed derivation of the spatial ground

motion power spectral density function and the ground motion parameters can be found in

[1]. Substructure method is used to analyse SSI effect. The dynamic impedances of the

foundations are defined in Section 4.3.1. The mean peak responses of the system are

estimated based on random vibration method after the power spectral densities of the

relative displacements are derived in Section 4.3.2.

4.3.1 Dynamic soil stiffness

As mentioned above, the soil-abutment interaction is ignored at sites 1 and 2, the only SSI

considered in the paper is the 2×2 pile group embedded in a uniform stratum at site 3.

The dynamic stiffness of a pile group ( GK ) can be calculated using the dynamic stiffness of

a single pile ( SK ) in conjunction with dynamic interaction factors ( 'α ). This method can

be used with confidence for pile groups not having a large number of piles, say less than 50

[18].

The dynamic stiffness and damping of a single pile can be described in terms of complex

stiffness

SSS cikK ω+= (4-1)

where superscript S represents the values for a single pile, ω is circular frequency, Sk and Sc are the stiffness and equivalent viscous damping of the pile. The corresponding

expressions for Sk and Sc can be readily obtained for use from the previous studies [28,

29], and the coefficients suggested by Gazetas [29] are used in the present paper.


4-8

In geotechnical practice, when the response of a pile group is of interest, such pile-soil-pile

interaction effects are often assessed through the use of an interaction factor 'α . For two

identical piles, the frequency-dependent dynamic interaction factor 'α is defined as

qq

qp

ww

== )('' ωαα (4-2)

where qpw is the dynamic displacement of pile q caused by pile p and qqw is the

displacement of pile q under own dynamic load. Gazetas et al. [30] presented the dynamic

interaction factors for floating pile groups in graphs; Dobry and Gazetas [31] developed a

simple analytical solution for computing the dynamic impedances of pile groups due to

pile-soil-pile interaction. The simple solution proposed by Dobry and Gazetas [31] is

adopted herein.

The frequency-dependent dynamic stiffness and damping coefficient of the pile group then

can be estimated as

( )Ghh Kk Re= , ( ) ω/Im G

hh Kc = (4-3a)

in the horizontal direction, and

( )Grr Kk Re= , ( ) ω/Im G

rr Kc = (4-3b)

in the rotational direction, where GhK and G

rK are the complex stiffness of the pile group

in the horizontal and rotational direction, respectively. Im and Re denote the real and

imaginary part of the pile group impedances.

4.3.2 Structural response formulation

With the hypotheses mentioned above, the total displacements of bridge decks and the pier

are

111 uuu gt += , 222 uuu g

t += , 3033 uLuuu gt +++= φ (4-4)


4-9

The dynamic equilibrium equations of the idealized model in Figure 4-1(b) can be

expressed in the matrix form as follows:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

++++−−++++−−++++−−

−−−−−−

+

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

++++−−++++−−++++−−

−−−−−−

+

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

3

2

1

2121

2121

2121

22

11

3

2

1

2121

2121

2121

22

11

3

2

1

3

3

3

2

1

3

2

1

221212121

21212121

21212121

2222

1111

3

2

1

221212121

21212121

21212121

2222

1111

3

2

1

333

333

333

2

1

00

00

000000

0000

/

2002

/

2002

000000

00000000

g

g

g

bbbb

bbbb

bbbb

bb

bb

g

g

g

bbbb

bbbb

bbbb

bb

bb

g

g

g

o

rbbbbbbbb

bbhbbbbbb

bbbbpbbbb

bbbb

bbbb

o

rbbbbbbbb

bbhbbbbbb

bbbbpbbbb

bbbb

bbbb

o

uuu

kkkkkkkkkkkk

kkkk

uuu

cccccccccccc

cccc

uuu

mmm

mm

Luuuu

Lkkkkkkkkkkkkkkkkkkkkkkkkkkk

kkkkkkkk

Luuuu

Lccccccccccccccccccccccccccc

cccccccc

Luuuu

mmmmmmmmm

mm

&

&

&

&&

&&

&&

&

&

&

&

&

&&

&&

&&

&&

&&

φ

φφ

(4-5)

We define the circular frequencies of the structure and the subsoil system in the following

form

1

121

2mkb=ω ,

2

222

2mkb=ω ,

321

23 mmm

k p

++=ω (4-6a)

321

2

mmmkh

h ++=ω , 2

321

2

)( Lmmmkr

r ++=ω (4-6b)

the corresponding viscous damping ratio of the bridge deck, pier and the subsoil system are

expressed as

1

111 2 b

b

kcωξ = ,

2

222 2 b

b

kcωξ = ,

p

p

kc

23

3

ωξ = (4-7a)

h

hhh k

c2ωξ = ,

r

rrr k

c2ωξ = (4-7b)

With the aid of Equations (4-6) and (4-7), and by assuming the mass ratios


4-10

3

1

mm

=α , 3

2

mm

=β (4-8)

Equation (4-5) can be expressed in the frequency domain as

{ } { })()]([)()]([ ωωωω iuiZiuiZ gg= (4-9)

where

{ } { }TiLiuiuiuiuiu )()()()()()( 0321 ωφωωωωω = (4-10a)

and

{ } { }Tgggg iuiuiuiu )()()()( 321 ωωωω = (4-10b)

are the dynamic response vector and the input ground motion vector, respectively. [ ])( ωiZ

and )]([ ωiZg are the impedance matrices of the system, which are in the following form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

)()(

)()()]([

5551

1511

ωω

ωωω

iziz

iziziZ

L

MM

L

(4-11a)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=)()(

)()()]([

5351

1311

ωω

ωωω

iziz

iziziZ

gg

gg

g

L

MM

L

(4-11b)

The elements of these two matrixes are given in Appendix A. The dynamic response of the

bridge structure can then be calculated by

{ } { } { })()]([)()]([)]([)( 1 ωωωωωω iuiHiuiZiZiu ggg == − (4-12)

For the bridge model shown in Figure 4-1, the minimum separation distance a MEJ must

provide to preclude pounding equals the relative displacement of the bridge system, which

can be expressed in the frequency domain as


4-11

)()( 11 ωω iui =Δ , )()( 22 ωω iui =Δ , )()()( 213 ωωω iuiui tt −=Δ (4-13)

The power spectral density functions of 1Δ , 2Δ and 3Δ thus can be derived, and the

corresponding expressions are given in Appendix B. It should be noted that Δ1 and Δ2, i.e.,

the relative displacement between bridge deck and abutments, only depend on the

respective dynamic response of the bridge deck, whereas the relative displacement response

between two adjacent decks, Δ3, depend on both bridge deck dynamic response and spatial

ground displacements.

After the derivation of the power spectral density functions of the required separation

distances, the mean peak responses can be estimated based on the standard random

vibration method [32]. In the present study, the high cut-off frequency is assumed to be 25

Hz since it covers the predominant vibration modes of most engineering structures and the

dominant earthquake ground motion frequencies.

4.4 Numerical example

This section carries out parametric studies of the effects of ground motion spatial variation,

site effect and SSI on the minimum separation gaps that the MEJs between bridge decks or

at abutments of the bridge shown in Figure 4-1 must provide. Three types of soil, i.e., firm,

medium and soft soil, are considered. Table 4-1 gives the corresponding parameters of soil

and base rock. Figure 4-2 shows the frequency-dependent dynamic stiffness and damping

coefficients of the pile group at site 3 corresponding to different soil properties. Because

this study concentrates on analysing SSI effect and the influences of varying site conditions

on bridge responses have been extensively studied in Reference [1], soil conditions at the

three sites are assumed to be the same in each case in the present study. To study the

ground motion spatial variation effect, highly, intermediately and weakly correlated ground

motions are studied. The corresponding parameters can be found in [1].


Type Density (kg/m3) Shear wave velocity (m/s) Damping ratio Poisson’s ratio

Base rock 3000 1500 0.05 0.4

Firm soil 2000 400 0.05 0.4

Medium soil 2000 200 0.05 0.4

Soft soil 1500 150 0.05 0.4


4-12

For two independent structures, the frequency ratio is usually used to measure the

vibration properties of the two adjacent structures, and it is found that the frequency ratio

has a great influence on the required separation distances between the two structures [1, 11,

24, 25]. In the present study, though the two adjacent bridge decks are coupled with each

other on the top of the pier, the uncoupled frequency ratio 12 / ff of the two spans is still

used to approximately quantify the frequency difference of the two decks. This is because

the uncoupled vibration frequency of each span is easy to calculate and has a

straightforward physical meaning. In this paper, the numerical results are presented with

respect to the dimensionless parameter 12 / ff . The stiffness of the left span is assumed to be

a constant with 71 104.2 ×=bk N/m, which gives an uncoupled frequency

0.12//2 111 == πmkf b Hz. The bearing stiffness of the right span varies from 5102×

N/m to 8105.1 × N/m to obtain different frequency ratios 12 / ff . As mentioned above, a

constant damping ratio of 5% is used for bearings and the pier, thus the damping

coefficient of the right span also varies with the stiffness to keep the damping ratio

unchanged in the calculations.

Figure 4-2. Frequency-dependent dynamic stiffness and damping coefficients of the pile

group (a)(b) horizontal direction and (c)(d) rotational direction

4.4.1 Influence of site effect and SSI


4-13

Figure 4-3 shows the influences of site effect and SSI on the total relative response. The

ground motions are assumed to be intermediately correlated and the apparent wave

velocity is 1000=appv m/s. Results corresponding to soft, medium and firm soils with SSI

(bold lines) or without SSI (thin lines) are compared, and the effect of SSI is examined.

Figure 4-3. Influence of site effect and SSI on the required separation distances

(a) 3Δ , (b) 2Δ and (c) 1Δ

As shown in Figure 4-3, for 3Δ and 2Δ the largest required separation distance occurs at

18.0/ 12 =ff , or when the uncoupled vibration frequency of the right span is 0.18 Hz.

This is because 0.18 Hz coincides with the central frequency of ground displacement as

shown in Figure 4-4(c). This significantly increases the displacement response of the right

span and hence the relative displacement response 3Δ and 2Δ .

When soft soil (bold solid line) is considered, additional amplification in 3Δ and especially

in 2Δ occurs at 75.0/ 12 =ff because the bridge system resonates with local site. At this

frequency ratio the vibration frequency of the right span is 0.75 Hz, which coincides with

the predominant frequency of the soft soil site with a depth of 50m and shear wave

velocity 150 m/s. This again amplifies the dynamic response of the right span and hence

increases the corresponding relative displacement. For medium soil (bold dash line), one

obvious peak occurs at 0.1/ 12 ≈ff in 2Δ owing to the same reason. However, this peak is

not observed in 3Δ , this is because 3Δ measures the relative displacement between the two

adjacent bridge decks, the two spans tend to vibrate in phase when their frequencies are

close to each other, therefore the relative displacement is the smallest although both

adjacent span displacements are large owing to resonance. When the site is relatively stiff

(dash dotted line), no obvious additional peak can be observed, because the fundamental


4-14

vibration frequency of the site is relatively high compared to the soft and medium site. In

this situation, the dynamic response of the right span is small even at resonance with the

site. The response of the stiff right span is primarily determined by the quasi-static

response associated with the non-uniform ground displacement at the multiple bridge

supports, and the quasi-static response is independent of the structural frequency, and is a

constant once the ground displacement is defined. Similar observation was also made in a

previous study [11].

As for the relative displacement 1Δ between the left abutment and the deck, although the

stiffness of the left span remains unchanged, 1Δ in Figure 4-3(c) is not a constant and

varies with the change of 2f because of the coupling through the pier. 1Δ experiences a

sudden significant variation when 12 / ff is close to unity. This is because changing 2f has

only insignificant influence on the vibration frequency of the left span when 2f differs

pronouncedly from 1f . However, when 2f is close to 1f , or 12 / ff is close to unity,

changing the vibration properties of the right span has a significant effect on the actual

vibration frequency of the left span through coupling at the pier. This results in a sudden

change in the response of the left span. Moreover, when right span resonates with the soil

site, it also more significantly affects the responses of the left span through coupling.

Therefore corresponding peak responses also occur at respective frequency ratios when the

right span resonates with the site. However, the peak response at 18.0/ 12 =ff is not

observed in 1Δ . This is because 1f is a constant in the simulations, and this peak is

associated primarily with the ground displacement, which has insignificant effect on the

dynamic responses of the left span through the dynamic coupling at the pier. The same

phenomenon was also observed in the previous study [1].

Larger 2Δ in Figure 4-3(b) is usually obtained when the site is soft. However, for 3Δ and

1Δ , medium soil (bold dash line) might give larger responses than the soft soil condition.

This is because the frequency of the left span is 1.0 Hz, which coincides with fundamental

frequency of the medium soil site with a depth of 50 m and shear wave velocity 200 m/s.

These observations indicate that the effect of local site conditions should not be neglected

because the structure might resonate with local site to therefore generate larger structural

responses, and hence larger separation distances will be required.


4-15

Compared with the results without SSI effect in Figure 4-3, SSI only slightly changes the

frequency content of the bridge, i.e., the peaks appear at almost the same frequency ratio.

This is because the power spectral densities of the required separation distances can be

expressed as the product of power spectral density of motions on the ground surface

( )( ωiS ) and the frequency response function of the structure ( )( ωiH ) as shown in

Appendix B. Local site amplifies certain frequencies significantly at various vibration

modes of the site, which results in the energy of the surface motion concentrates at a few

frequencies. Take soft site for example, at the site fundamental frequency of 0.75 Hz the

soft soil amplifies ground motions on the base rock 24 times as shown in Figure 4-4(a),

which alters the ground motions on ground surface significantly from the base rock motion

as can be seen in Figure 4-4(b). The influence of site effect is more significant than that of

the frequency response function. That is why larger relative displacement occurs at the

same frequency ratio of the bridge spans with or without considering the SSI effect because

this frequency ratio corresponds to the resonance of either one bridge span with the site.

Figure 4-4. Site effect on ground motion spatial variations: (a) transfer function,

(b) PSD of surface acceleration and (c) PSD of surface displacement

To observe the contribution of SSI more clearly, the required separation distances with

consideration of SSI are subtracted by those without SSI effect, and the results are shown

in Figure 4-5. As expected the influence of SSI is significant especially for soft and medium

soil site condition. The required separation distances will be underestimated when SSI

effect is ignored since most of the values shown in Figure 4-5 are larger than zero. As

shown in Figure 4-5(b), when soft or medium soil is considered, the influence of SSI on

2Δ increases and becomes most pronounced when the structure resonates with local site.

The contribution of SSI on relative displacement response is nearly 0.2 m for soft soil

when 12 / ff is around 0.75. As previously discussed, the right span resonates with local site

at this frequency ratio. When 75.0/ 12 >ff , the influence of SSI on the total responses


4-16

decreases and becomes almost a constant when the right span is stiff enough, indicating

quasi-static response dominates the total response. It is generally true that SSI effect is

more obvious on a soft soil site than on a medium soil site. When a firm site is considered

the influence of SSI can be neglected. For 3Δ , similar observations of different SSI effect

can be obtained. It should be noted that no obvious peak can be observed when resonance

occurs for medium soil, owing to the two spans tend to vibrate in phase as discussed

above. For 1Δ , it is observed again that SSI effect on soft site is more prominent than on

firm site.


conditions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ


conditions ( 0.21 =f Hz ) (a) 3Δ , (b) 2Δ and (c) 1Δ

Reference [1] concluded that considering only the frequency ratio of the adjacent spans is

not enough to determine the required separation distances. The absolute frequency of the

bridge also strongly affects the responses. The same fact is expected in this study when SSI

effect is considered. To cover a wide range of possible cases, a relatively stiff left span with

0.21 =f Hz is additionally investigated. Figure 4-6 shows the contribution of SSI to the

total responses. For 3Δ and 2Δ , similar conclusions can be obtained as in Figure 4-5, i.e.,


4-17

the contribution of SSI of a soft soil site is generally larger than that of a firm soil site and

is most significant when resonance occurs. For 1Δ , however, the influence of site

conditions is less prominent as compared to the case with 0.11 =f Hz, the influence of SSI

can actually be neglected. This is because the left span is relatively rigid, which reduces the

SSI effect. Moreover, the right spans are only weakly coupled through the bearings at the

pier cap, and the coupling effect is less pronounced when the left span bearings are stiff.

4.4.2 Influence of ground motion spatial variation and SSI

To study the influences of ground motion spatial variation and SSI effect on the required

separation distances, highly, intermediately and weakly correlated ground motions are

considered. The soils under the three foundations are assumed to be medium soil. The

apparent wave velocity is 1000=appv m/s. As shown in Figure 4-7, whether or not SSI is

considered, the influence pattern of ground motion spatial variation on the required

separation distances does not change too much. Take 3Δ in Figure 4-7(a) for example,

when SSI is not considered, ground motion spatial variation effect is most significant when

12 / ff is close to unity, where weakly correlated ground motions result in larger required

separation distance. The ground motion spatial variation effect is, however, not so

pronounced if the vibration frequencies of the two spans differ significantly. In these

situations, the out-of-phase vibration of the two spans owing to their different frequencies

contributes most to the relative displacement of the adjacent bridge decks. When SSI is

included, similar results can be observed. Similar observations can be made on 2Δ and 1Δ .

These observations are also similar to those reported in [1] where SSI is not considered. As

shown, the influence of changing spatial ground motions from highly correlated to weakly

correlated has insignificant effect on the relative displacement responses. This is because

the local site effects dominate the structural responses, as discussed above. Under a

uniform site condition at the multiple bridge supports, the effect of cross correlation

between spatial ground motions will be more prominent as observed in many previous

studies (e.g., [11]).


4-18

Figure 4-7. Influence of ground motion characteristics and SSI on the required separation

distances ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ

To see the influence of SSI more clearly, Figures 4-8 and 4-9 show the subtracted

displacements between the results with and without consideration of SSI effect for soft

( 0.11 =f Hz) and stiff ( 0.21 =f Hz) left span bearings, respectively. It is obvious in Figure

4-9 that the effect of SSI is most significant at 5.0/ 12 ≈ff , or 0.12 =f . As discussed

above, natural vibration frequency of the 50 m medium site is about 1.0 Hz. At this

frequency ratio, the right span resonates with the local site, which results in large responses.

Large right span response also affects the response of the left span through the coupling at

the pier, but at a less scale. Figure 4-8, however, does not show the same response

phenomenon. For 2Δ in Figure 4-8(b), similar observation can be obtained when right

span resonates with the site at 0.1/ 12 ≈ff or 0.12 =f . For 3Δ and 1Δ in Figure 4-8(a) and

(c), however, no such peaks can be observed. This is because both the left and right spans

resonate with the site at 0.1/ 12 ≈ff . Since the two adjacent spans tend to vibrate in-phase

at this vibration frequency, the relative displacement response 3Δ is the smallest, and the

effect of SSI is also insignificant. As for 1Δ , it shows a fluctuation when 12 / ff is close to

unity as discussed in the previous section.

It is interesting to observe that with an increase in the spatial variability of the ground

motion, the required separation distance becomes less sensitive to the dynamic interaction,

i.e., SSI effect becomes more pronounced when the spatial ground motions have higher

correlations, whereas the SSI effect is less prominent if the spatial ground motion is less

correlated. These results are in agreement with that of Shrikhande and Gupta [21], where

they investigated the influences of ground motion spatial variation and SSI on the bending

moment at the mid-point of the centre span of a suspension bridge based on the stochastic

approach in the frequency domain. These observations are, however, not fully consistent


4-19

with those in [24], in which it was concluded that SSI effect is more prominent only when

the spatial ground motions are highly correlated in the range of 12 / ff between 0.55 and

1.0. The later study [24] was carried out in the time and Laplace domain by using 20 sets of

stochastically simulated time histories as inputs, which may give biased results because of

the limited number of simulations. The results obtained in this study demonstrate the

importance of considering SSI effect, especially when the spatial ground motions are highly

correlated.


coherency loss functions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ


coherency loss functions ( 0.21 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ

4.5 Conclusions

Based on the fixed base assumption, Reference [1] investigated the combined effect of

ground motion spatial variation and site effect on the required separation distances that

modular expansion joints (MEJs) must provide to prevent seismic pounding. This paper is

an extension of Reference [1] by including the SSI effect using the substructure method.

The soil surrounding the pile foundation is modelled by equivalent frequency-dependent

horizontal and rotational spring-dashpot systems. The combined effect of site condition


4-20

and SSI, combined effect of ground motion spatial variation and SSI are investigated, and

the SSI effect is highlighted. Following conclusions are obtained:

1. The influence of SSI on the required separation distances is significant. Larger

separation distances to avoid seismic pounding are usually required when SSI is

considered.

2. SSI effect can not be neglected especially when the structures are founded on soft

site. The contribution of SSI is relatively small when firm site is considered.

3. SSI effect is most evident when the structure resonates with local site.

4. SSI effect on the required separation distances is more prominent when the spatial

ground motions are highly correlated. Otherwise ground motion spatial variation

effect is more pronounced. Local site conditions are always important and should

not be neglected for an accurate structural response analysis.

4.6 Appendix

4.6.1 Appendix A: Element for [ ])( ωiZ and )]([ ωiZg

1121

211 2)( ξωωωωω iiz ++−=

0)(12 =ωiz 2/)2()( 11

2113 ξωωωω iiz +−= (A4-1)

2/)2()( 112114 ξωωωω iiz +−=

2/)2()( 112115 ξωωωω iiz +−=

0)(21 =ωiz

2222

222 2)( ξωωωωω iiz ++−=

2/)2()( 222223 ξωωωω iiz +−= (A4-2)

2/)2()( 222224 ξωωωω iiz +−=

2/)2()( 222225 ξωωωω iiz +−=

2/)2()( 11

2131 ξωωωαω iiz +−=

2/)2()( 222232 ξωωωβω iiz +−=

)2)(1(2/)2(2/)2()( 332322

2211

21

233 ξωωωβαξωωωβξωωωαωω iiiiz ++++++++−=

2/)2(2/)2()( 222211

21

234 ξωωωβξωωωαωω iiiz ++++−= (A4-3)

2/)2(2/)2()( 222211

21

235 ξωωωβξωωωαωω iiiz ++++−=


4-21

2/)2()( 112141 ξωωωαω iiz +−=

2/)2()( 222242 ξωωωβω iiz +−=

2/)2(2/)2()( 222211

21


)2)(1(2/)2(2/)2()( 222

2211

21

244 hhh iiiiz ξωωωβαξωωωβξωωωαωω ++++++++−=

2/)2(2/)2()( 222211

21


2/)2()( 11

2151 ξωωωαω iiz +−=

2/)2()( 222252 ξωωωβω iiz +−=

2/)2(2/)2()( 222211

21


2/)2(2/)2()( 222211

21


)2)(1(2/)2(2/)2()( 222

2211

21

255 rrr iiiiz ξωωωβαξωωωβξωωωαωω ++++++++−=

2/)2()( 11

21

211 ξωωωωω iizg +−=

0)(12 =ωizg (A4-6)

2/)2()( 112113 ξωωωω iizg +=

0)(21 =ωizg

2/)2()( 2222

222 ξωωωωω iizg +−= (A4-7)

2/)2()( 222223 ξωωωω iizg +=

2/)2()( 11

2131 ξωωωαω iizg +=

2/)2()( 222232 ξωωωβω iizg += (A4-8)

222

2211

2133 2/)2(2/)2()( ωξωωωβξωωωαω ++−+−= iiizg

2/)2()( 11


2/)2()( 222242 ξωωωβω iizg += (A4-9)

222

2211


2/)2()( 11


2/)2()( 222252 ξωωωβω iizg += (A4-10)

222

2211



4-22

4.6.2 Appendix B: PSDs of the required separation distances

[ ]

[ ] [ ])()()(Re2)()()(Re2

)()()(Re2)()(1

)()(1)()(1)(

23131241313114

1212114332

134

222

124112

1141

ωωωω

ωωωω

ωωωω

ωωω

ωωω

ωωω

ω

iSiHiHiSiHiH

iSiHiHiSiH

iSiHiSiHS

∗∗

∗

Δ

++

++

+=

(B4-1)

[ ]

[ ] [ ])()()(Re2)()()(Re2

)()()(Re2)()(1

)()(1)()(1)(

23232241323214

1222214332

234

222

224112

2142

ωωωω

ωωωω

ωωωω

ωωω

ωωω

ωωω

ω

iSiHiHiSiHiH

iSiHiHiSiH

iSiHSiHS

∗∗

∗

Δ

++

++

+=

(B4-2)

[ ][ ]{ }[ ][ ]{ }[ ][ ]{ })()()(1)()(Re2

)()()(1)()(Re2

)(1)()(1)()(Re2

)()()(1

)(1)()(1

)(1)()(1)(

23231322124

13231321114

12221221114

332

23134

222

22124

112

211143

ωωωωωω

ωωωωωω

ωωωωωω

ωωωω

ωωωω

ωωωω

ω

iSiHiHiHiH

iSiHiHiHiH

iSiHiHiHiH

iSiHiH

iSiHiH

iSiHiHS

∗

∗

∗

Δ

−−−+

−+−+

−−+−+

−+

−−+

+−=

(B4-3)

where )( ωiS jk 3,2,1, =kj is the cross power spectral density function between points j

and k on the ground surface, which can be obtained from Reference [1]. )( ωiH jk is the

frequency response function, and can be determined by Equation (4-12).

4.7 References




2. Hall FJ, editor. Northridge earthquake, January 17, 1994. Earthquake Engineering

Research Institute, Preliminary reconnaissance report, EERI-94-01; 1994.


4-23



Engineering JSCE 1996; 13(2): 211-240.




2006. Mid-America Earthquake Centre. Report No. 07-02, 57, 2007.


earthquake on bridges. The 14th World Conference on Earthquake Engineering. Beijing,

China, 2008; S31-006.



1538.


response of multi-frame bridges. Journal of Structural Engineering ASCE 2002; 128(7):

860-869.



1998; 27(5): 487-502.

10. Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic




earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1): 91-103.






14. Dumanogluid, AA, Soyluk K. A stochastic analysis of long span structures

subjected to spatially varying ground motions including the site-response effect.


15. Ates S, Bayraktar A, Dumanogluid, AA. The effect of spatially varying earthquake

ground motions on the stochastic response of bridges isolated with friction

pendulum systems. Soil Dynamics and Earthquake Engineering 2006; 26: 31-44.

16. Wolf JP. Dynamic soil-structure interaction. Englewood Cliffs, NJ: Prentice Hall; 1985.


4-24

17. Spyrakos CC, Vlassis AG. Effect of soil-structure interaction on seismically isolated

beiges. Journal of Earthquake Engineering 2002; 6(3): 391-429.

18. Makris N, Badoni D, Delis E, Gazetas G. Prediction of observed bridge response

with soil-pile-structure interaction. Journal of Structural Engineering 1994; 120(10):

2992-3011.

19. Chouw N, Hao H. Study of SSI and non-uniform ground motion effect on

pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005; 25:

717-728.




21. Shrikhande M, Gupta VK. Dynamic soil-structure interaction effects on the seismic

response of suspension bridges. Earthquake Engineering and Structural Dynamics 1999;

28(11): 1383-1403.

22. Sextos AG, Pitilakis KD, Kappos AJ. Inelastic dynamic analysis of RC bridges


interaction phenomena. Part 1: Methodology and analytical tools. Earthquake


23. Sextos AG, Pitilakis KD, Kappos AJ. Inelastic dynamic analysis of RC bridges


interaction phenomena. Part 2: Parametric study. Earthquake Engineering and




Structures 2008; 30(1): 154-162.

25. Chouw N, Hao H. Seismic design of bridge structures with allowance for large

relative girder movements to avoid pounding. New Zealand Society for Earthquake

Engineering Conference. Wairakei, New Zealand 2008; Paper No: 10.

26. Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soil-structure

interaction. Soil Dynamics and Earthquake Engineering 2003; 23: 287-302.

27. Fan K, Gazetas G, Kaynis A, Kausel E, Ahmad S. Kinematic seismic response of

single piles and pile groups. Journal of Geotechnique Engineering 1991; 117(12): 1860-

1879.

28. Novak M, Sharnouby BL. Stiffness constants of single piles. Journal of Geotechnical

Engineering 1983; 109(7): 961-974.


4-25

29. Gazetas G. In: Fang HY, editor. Foundation vibrations, foundation engineering handbook,

2nd edition 1991; 553-593.

30. Gazetas G, Fan K, Kaynia A, Kausel E. Dynamic interaction factors for floating

pile groups. Geotechnical Engineering 1991; 117(10): 1531-1548.

31. Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating

pile groups. Geotechinique 1988; 38(4): 557-574.


Mechanics 1980; 106(6): 1195-1213.


5-1

Chapter 5 Modelling and simulation of spatially varying earthquake ground motions at a canyon site with multiple soil layers

By: Kaiming Bi and Hong Hao

Abstract: In a flat and uniform site, it is reasonable to assume the spatially varying

earthquake ground motions at various locations have the same power spectral density or

response spectrum. If a canyon site is considered, this assumption is no longer valid

because of different local site amplification effect. This paper models and simulates

spatially varying ground motions on surface of a canyon site in two steps. In the first step,

the base rock motions at different locations are assumed to have the same intensity, and are

modelled by a filtered Tajimi-Kanai power spectral density function or other stochastic

ground motion attenuation models. The ground motion spatial variation is modelled by an

empirical coherency loss function. The power spectral density functions of the surface

motions on the canyon site with multiple soil layers are derived based on the deterministic

wave propagation theory, assuming the base rock motions consist of out-of-plane SH wave

or in-plane combined P and SV waves propagating into the site with an assumed incident

angle. In the second step, a stochastic method to generate spatially varying time histories

compatible with non-uniform spectral densities and a coherency loss function is developed

to generate ground motion time histories on a canyon site. Two numerical examples are

presented to demonstrate the proposed method. Each generated ground motion time

history is compatible with the derived power spectral density at a particular point on the

canyon site or response spectrum corresponding to the respective site conditions, and any

two of them are compatible with a model coherency loss function.

Keywords: ground motion simulation; spectral representation method; wave propagation

theory; power spectral density function; response spectrum


5-2

5.1 Introduction

In the design of structures to resist strong earthquake ground excitations, properly define

ground motions is crucial for a reliable analysis of structural responses. Besides ground

motion time histories, ground motion response spectrum and power spectral density

function are the most commonly used parameters to define earthquake action. For the

‘point’ structures, owing to the dimensions of such structures are relatively small compared

to the wavelengths of the seismic motions, it is reasonable to assume the ground motions

over the entire structural base are the same. Many ground motion power spectral density

functions have been developed by different researchers, e.g., the Tajimi-Kanai power

spectrum model [1] and the Clough-Penzien model [2]. Both of them were proposed by

assuming the base rock excitation was a white noise random process, and the surface

ground motion was estimated by calculating the responses of a single soil layer to the white

noise excitation. Many stochastic ground motion models [3-5] have also been proposed by

considering the rupture mechanism of the fault and the path effect for transmission of

waves through the media from the fault to the ground surface. Local site effect, which

amplifies and filters the incoming seismic waves and hence changes their amplitudes and

the frequency contents, has also been intensively studied by many researchers. Wolf [6]

presented a uniform approach in the frequency domain to analyze both the free-field

response and the soil-structure interaction effect based on the wave propagation theory and

finite element approach. Wolf [7] and Safak [8] also presented methods to model the

propagating shear waves in layered media in the time domain.

For large dimensional structures, such as long span bridges, pipelines, communication

transmission systems, the ground motions at different stations during an earthquake are

inevitably different, which is known as the ground motion spatial variation effect. There

are many reasons that may result in the spatial variability in seismic ground motions, e.g.,

the wave passage effect owing to the different arrival times of waves at different locations;

the loss of coherency due to seismic waves scattering in the heterogeneous medium of the

ground; the site amplification effect owing to different local soil properties. It has been

proved that ground motion spatial variations have great influence on the structural

responses and in some cases might even govern the structural responses. The ground

motion spatial variations are usually modelled by a theoretical/semi-empirical power

spectral density function and a coherency loss function. Many ground motion spatial

variation models have been proposed especially after the installation of the SMART-1 array

in Lotung, Taiwan. Zerva and Zervas [9] overviewed these models. It should be noted that


5-3

most of these models were proposed based on the seismic data recorded from the relatively

flat-lying sites. Taking different soil conditions into consideration, Der Kiureghian [10]

proposed a theoretical coherency loss function, in which the ground motion power spectral

density function was represented by a site-dependent transfer function and a white noise

spectrum. Typical site-dependent parameters, i.e., the central frequency and damping ratio

for three generic site conditions, namely, firm, medium and soft site were proposed. The

advantage of the model is that it can consider different soil properties at different support

locations and it is straightforward to use. The drawback is that it can only approximately

represent the local site effects on ground motions. For example, it is well known that

seismic wave will be amplified and filtered when propagating through a layered soil site.

The amplifications occur at various vibration modes of the site. Therefore, the energy of

surface motions will concentrate at a few frequencies. The power spectral density function

of the surface motion then may have multiple peaks. This phenomenon, however, cannot

be considered in Der Kiureghian’s model since only one peak corresponding to the

fundamental vibration mode of the site is modelled.

The ground motion power spectral density functions and spatial variation models can be

used directly as inputs at multiple supports of structures in spectral analysis of structural

responses. This approach, however, is usually applied to relatively simple structural models

and for linear response of the structures owing to its complexity. For complex structural

systems and for nonlinear seismic response analysis, only the deterministic solution can be

evaluated with sufficient accuracy. In this case, the generation of artificial seismic ground

motions is required. Many methods are also available to generate artificial spatially

correlated time histories at different structural supports. Hao et al. [11] presented a method

of generating spatially varying time histories at different locations on ground surface based

on the assumption that all the spatially varying ground motions have the same intensity, i.e.,

the same power spectral density or response spectrum. The variation of the spatial ground

motions is modelled by an empirical coherency loss function and a phase delay depending

on a constant apparent wave propagation velocity. If the considered site is flat with

uniform soil properties, the uniform ground motion intensity assumption for spatial

ground motions in the site is reasonable. However, for a canyon site or a site with varying

soil properties, because local site conditions affect the wave propagation hence the ground

motion intensity and frequency contents, the uniform ground motion power spectral

density assumption is no longer valid. Deodatis [12] developed a method to simulate spatial

ground motions with different power spectral densities at different locations. The method

is based on a spectral representation algorithm [13, 14] to generate sample functions of a


5-4

non-stationary, multivariate stochastic process with evolutionary power spectrum. Similar

to the Der Kiureghian [10] model, the considered varying spectral densities are filtered

white noise functions with different central frequency and damping ratio. This method thus

can only approximately represent local site effects on ground motions as discussed above.

Moreover, trying to establish an analytical expression for a realistic ground motion

evolutionary power spectrum related to the local site conditions is quite difficult since very

limited information is available on the spectral characteristics of propagating seismic waves

[15].

On the other hand, many studies of site amplifications of seismic waves have also been

reported. Taking the site amplification effect into consideration, Hao [16] developed a

numerical method to calculate the site amplification effects on ground motion time

histories by assuming seismic waves consisting of SH, and combined P and SV waves.

Wang and Hao [17] then further extended this method to include the effects of random

variation of soil properties on site amplifications of seismic waves. Some computer

programs, such as SHAKE [18], EERA [19] and NERA [20], are also available to calculate

site responses to incoming seismic waves and hence the ground motion time histories on

the ground surface by solving the fundamental dynamic equations of motion in the

frequency domain. It should be noted that these approaches only simulate ground motion

time histories at one point on ground surface, ground motion spatial variations are not

considered. Studies which consider both the ground motion spatial variation effect and the

site amplification effect are limited.

This paper combines the wave propagation theory [6] and the spectral representation

method [13, 14] to derive the power spectral density functions of the spatially varying

ground motion on surface of a canyon site with multiple soil layers. The ground motion

spatial variations are modelled in two steps: firstly, the spatially varying base rock ground

motions are assumed to consist of out-of-plane SH wave or in-plane combined P and SV

waves and propagate into the layered soil site with an assumed incident angle. The spatial

base rock motions are assumed to have the same intensity and frequency contents and are

modelled by the filtered Tajimi-Kanai power spectral density function [2]. The spatial

variation effect is modelled by an empirical coherency loss function [21]. The surface

motions of a canyon site with multiple soil layers are derived based on the deterministic

wave propagation theory. The auto power spectral density functions of ground motions at

various points on ground surface and the cross power spectral density functions between

ground motions at any two points are derived. The spectral representation method is then


5-5

used to generate spatially varying ground motion time histories compatible to the derived

auto and cross power spectral density functions. Compared to the work by Deodatis [12],

in this study the power spectral density functions at different locations of a canyon site are

derived based on the wave propagation theory, which directly relates the local soil

conditions and base rock motion characteristics with the surface ground motions, thus

local site effect can be realistically considered. Besides the filtered Tajimi-Kanai power

spectral density functions used in this study, other stochastic ground motion attenuation

models for different regions can be straightforwardly used to model base rock motion. The

current approach also allows for a consideration of different incoming wave types and

incident angles to the soil site, which have great influence on the surface motions. For the

completeness of the study, the ground motion time histories at a site with different soil

properties represented by different response spectra is also considered in the paper, and a

numerical example is given. The proposed approach can be used to simulate ground

motion time histories at an uneven site with known non-uniform site conditions. The

simulated time histories can be used as inputs to long-span structures with multiple

supports resting on site of varying conditions.

5.2 Wave propagation theory and site amplification effect

The one-dimensional (1D) wave propagation theory proposed by Wolf [6] is adopted in the

present study to consider the influence of local site effect. It should be noted that the

seismic waves are assumed to incident with an angle to the base rock and soil layer

interface, and then propagate vertically in the soil layers in the 1D wave propagation

theory, the scattering and diffraction of waves by canyons, which is a 2D wave propagation

problem, are not considered. Further studies are needed to incorporate the scattering and

diffraction effect into the simulation technique. For completeness, the 1D wave

propagation theory is briefly introduced here. More detailed information can be found in

Reference [6].

For a harmonic excitation with frequencyω , the dynamic equilibrium equations can be

written as

ec

ep2

22 ω

−=∇ or { } { }Ω−=Ω∇ 2

22

scω (5-1)


5-6

where e2∇ and { }Ω∇2 are the Laplace operator of the volumetric strain amplitude e and

rotational-strain-vector { }Ω . pc and sc are the P- and S-wave velocity, respectively. This

equation can be solved by using the P- and S-wave trial function. The out-of-plane

displacements with the amplitude v is caused by the incident SH wave, while the in-plane

displacements with the amplitude u and w in the horizontal and vertical directions depend

on the combined P and SV waves. The amplitude v is independent of u and w, hence, the

two-dimensional dynamic stiffness matrix of each soil layer for the out-of-plane and in-

plane motion, ][ LSHS and ][ L

SVPS − , can be formulated independently by analysing the

relations of shear stresses and displacements at the boundary of each soil layer. Assembling

the matrices of each soil layer and the base rock, the dynamic stiffness of the total system is

obtained and denoted by ][ SHS and ][ SVPS − , respectively. The dynamic equilibrium

equation of the site in the frequency domain is thus

{ } { }SHSHSH PuS =][ or { } { }SVPSVPSVP PuS −−− =][ (5-2)

where { }SHu and { }SHP are the out-of-plane displacements and load vector corresponding

to the incident SH wave, { }SVPu − and { }SVPP − are the in-plane displacements and load

vector of the combined P and SV waves. The stiffness matrices ][ SHS and ][ SVPS − depend

on soil properties, incident wave type, incident angle and circular frequency ω . The

dynamic load { }SHP and { }SVPP − depend on the base rock properties, incident wave type,

incident wave frequency and amplitude. By solving Equation (5-2) in the frequency domain

at every discrete frequency, the relationship of the amplitudes between the base rock and

each soil layer can be formed, and the site transfer function )]([ ωH at each soil layer can be

estimated. In the present study, only the motion on the ground surface is of interest.

To illustrate Equation (5-2), a site consisting of a single homogeneous layer resting on a

half-space is used as an example. The input on the base rock is assumed to be SH wave. It

can be directly extended to more soil layers or combined P and SV waves. Assembling the

dynamic stiffness matrix of the layer ][ LSHS and of the half-space R

SHS , the stiffness matrices

][ SHS , out-of-plane displacements { }SHu and load vector { }SHP can be expressed as [6]


5-7

{ }

{ } TRRSH

TbtSH

LL

L

L

LL

SH

vGiktP

vvu

dktipdktdkt

dktGktS

],0[

],[

sincos11cos

sin][

0*

*

=

=

⎥⎦

⎤⎢⎣

⎡

+−−

=

(5-3)

where k is the wave number, t is a parameter related to the incident angle, G is the shear

modulus, d is the depth of the soil layer, superscript R and L represents base rock and soil

layer respectively, vt and vb are the displacement at the top and bottom of the soil layer, T

denotes transpose, and i is the unit imaginary number.

The above formulation can be easily extended to more soil layers by assembling the proper

layer stiffness to the stiffness matrix. More detailed information for a multiple-layer site and

for the case with combined P and SV wave can be found in Reference [6].

Substituting Equation (5-3) into Equation (5-2), the ratio of surface motion tv to

outcropping motion 0v is

dktpidktv

vHLL

t

sincos

1)(0 +==ω (5-4)

in which LLGR GtGtp ** /= is the impedance ratio.

Considering linear elastic response only, the auto power spectral density functions of

ground motions at various points on ground surface and the cross power spectral density

functions between ground motions at any two points can be derived as

njiidSiHiHiS

niSiHS

jijigjiij

giii

,...,2,1,),()()()()(

,...,2,1)()()(

''''*

2

==

==

ωγωωωω

ωωω (5-5)

where )( ωiHi , )( ωiH j are the site transfer function at support i and j, respectively;

superscript ‘*’ denotes complex conjugate; )(ωgS is the ground motion power spectral


5-8

density at the base rock; ),( '''' ωγ idjiji

is the coherency loss function of spatial ground

motions at the base rock, which is related to the distance between location i’ and j’ directly

underneath the point i and j on ground surface as illustrated in Figure 5-1.

5.3 Ground motion simulation

Spatial earthquake ground motions on the base rock are assumed as stationary random

processes with zero mean values and having the same power spectral density function. This

is a reasonable assumption since the distance from the source to the site is usually much

larger than the dimension of the structure. The cross power spectral density function of

ground motions at n locations in a site can be written as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅

=

)()()(

)()()()()()(

)(

21

22221

11211

ωωω

ωωωωωω

ω

nnnn

n

n

SiSiS

iSSiSiSiSS

iS (5-6)

where )(ωiiS and njiiSij ,...,2,1,),( =ω are the auto and cross power spectral density

function respectively, defined in Equation (5-5).

The matrix )( ωiS is Hermitian and positive definite, it can be decomposed into the

multiplication of a complex lower triangular matrix )( ωiL and its Hermitian )( ωiLH :

)()()( ωωω iLiLiS H= (5-7)

The decomposition can be performed by using the Cholesky’s method. The lower

triangular matrix )( ωiL is in the following form:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅

=

)()()(

0)()(00)(

)(

21

2221

11

ωωω

ωωω

ω

nnnn LiLiL

LiLL

iL (5-8)

and


5-9

ijS

iSiSiSiL

niiSiSSL

jj

i

kjkikij

ij

i

kikikiiii

,...,2,1)(

)()()()(

,...,2,1)()()()(

1

1

*

2/11

1

*

=−

=

=⎥⎦

⎤⎢⎣

⎡−=

∑

∑

−

=

−

=

ω

ωωωω

ωωωω

(5-9)

After obtaining )( ωiL , the stationary time series nitui ,...,2,1),( = , can be simulated in the

time domain as [11]

)]()(cos[)()(1 1

nmnnimnn

i

m

N

nimi tAtu ωϕωβωω ++=∑∑

= =

(5-10)

where

Nim

imim

Nimim

iLiL

iLA

ωωωωωβ

ωωωωω

≤≤=

≤≤Δ=

− 0),)](Re[)](Im[(tan)(

0,)(4)(

1

(5-11)

are the amplitudes and phase angles of the simulated time histories which ensure the

spectrum of the simulated time histories compatible with those given in Equation (5-6);

)( nmn ωϕ is the random phase angles uniformly distributed over the range of ]2,0[ π , mnϕ

and rsϕ should be statistically independent unless rm = and sn = ; Nω represents an

upper cut-off frequency beyond which the elements of the cross power spectral density

matrix given in Equation (5-6) is assumed to be zero; ωΔ is the resolution in the

frequency domain, and ωω Δ= nn is the nth discrete frequency.

Directly use Equation (5-10) to generate ground motion is quite time consuming. Ground

motions can be generated more efficiently in the frequency domain based on the fast

Fourier transform (FFT) technique. The Fourier transform of )(tui is in the following

form [11]

NniBiU nimnimn

i

mimni ,...,2,1)],(sin)()[cos()(

1

=+= ∑=

ωαωαωω (5-12)


5-10

where )( nimB ω is the amplitude at frequency nω , and )( nim ωα is the corresponding phase

angle, defined by:

)()()(

2/)()(

nmnnimnim

nimnim AB

ωϕωβωα

ωω

+=

= (5-13)

The corresponding time series )(tui can be obtained by inverse transforming )( ni iU ω into

the time domain.

The time series generated by Equation (5-10) or (5-12) are stationary processes. In order to

obtain the non-stationary time histories, an envelope function )(tζ is applied to )(tui , the

non-stationary time histories at different locations are obtained by

nituttf ii ,...,2,1),()()( == ζ (5-14)

It should be noted that if the local site effect is not considered, the cross-power spectral

density functions given in Equation (5-5) become

njiidSiSjijigij ,...,2,1,),,()()( '''' == ωγωω (5-15)

This is because 1)()( == ωω iHiH ji when the site amplification effect is not considered. In

this case the spatial ground motions will have the same power spectral density function

Sg(ω), the spatial variation is modelled by the coherency loss function only. Then the above

approach is the same as that proposed by Hao et al. [11]. In other words, it is a special case

of the present study.

In engineering practice, design response spectrum for a given site is more commonly

available instead of the ground motion power spectral density function. Therefore it will be

very useful to generate ground motion time histories that are compatible to the given

design response spectrum. In previous works of generating spatially varying ground motion

time histories [11, 12], this is achieved by two steps. First the spatially varying ground

motion time histories are generated using an arbitrary power spectral density function, and

then adjusted through iterations to match the target response spectrum. Usually a few

iterations are needed to achieve a reasonably good match [11, 12]. In this paper, a similar


5-11

approach is used. However, the ground motion power spectral densities that are related to

the target design response spectra are derived first. The time histories are generated to be

compatible with these power spectral densities. With this approach, the iterations might not

be necessary for the simulated spatially varying time histories to be compatible to the

multiple target response spectra. Even if iterations are needed, the response spectra of the

simulated time histories converge to the target spectra faster. Therefore the current

approach is computationally more efficient. The method proposed in this paper is

introduced in the following.

For a given acceleration response spectrum )(ωRSA , the corresponding power spectral

density )(ωS can be estimated by [22]

)lnln(/)()( 2 pT

RSASωπω

πωξω −−= (5-16)

where ξ is the damping ratio, T the time duration and p the probability coefficient,

usually 85.0≥p [22].

Using the above approach, the generated time histories usually match well with the multiple

target response spectra. If the response spectra of the generated time histories )()( ωifRSA

do not match satisfactorily the target spectra, iterations need be carried out by adjusting the

power spectral density function, which is done by multiplying )(ωS by the ratio

2)( )](/)([ ωω ifRSARSA , and perform the simulation again. This process can be repeated

until satisfactory compatibility is achieved. Usually after 3 or 4 iterations, good match can

be obtained, as compared to the method by Deodatis [12], in which good match can be

obtained usually only after more than 10 iterations.

5.4 Numerical examples

An alluvium canyon site with multiple soil layers shown in Figure 5-1 is selected as an

example, in which h is the layer depth, G is the shear modulus, ρ density, ξ damping

ratio, υ Poisson’s ratio and α incident angle. Ground motions on the base rock and on

ground surface at three different locations indicated in the figure will be simulated in the

study.


5-12

Figure 5-1. A canyon site with multiple soil layers (not to scale)

5.4.1 Amplification spectra

The site amplification effect is studied first. For conciseness, only the amplification spectra

at site 3 are plotted. Figure 5-2(a) shows the amplification spectra for the horizontal out-of-

plane motions when SH wave propagates into the site with different incident angles. Figure

5-2(b) and Figure 5-2(c) show the amplification spectra for the in-plane horizontal and

vertical motions with an assumption that the incoming waves consist of combined P and

SV waves and the amplitude of the vertical motion is 2/3 of that of the horizontal

component.

As shown in Figure 5-2, different incoming waves and incident angles significantly affect

the site amplification spectra hence the surface motions in each direction. The site

amplifies the incident waves at various frequencies corresponding to respective vibration

modes of local site. Thus, the motions on the ground surface, which can be obtained by

multiplying the power spectral density function of the base rock motion with the site

amplification spectra at each discrete frequency, strongly depend on the local site

conditions. Moreover, the ground motion power spectral density functions may consist of

multiple distinctive peaks associated with the multiple modes of the site. The commonly

used filtered white noise is not able to represent the ground motion power spectral density

with multiple peaks. Table 5-1 gives the first two horizontal and vertical vibration

frequencies of the site.

3

No.1 Sandy fill, h=5m, G=30MPa, 3/1900 mkg=ρ , %5=ξ , 45.0=υ

1

2

1’ 2’ 3'

No.2 Soft Clay, h=15m, G=20MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ

No.3 Silt sand, h=6m, G=220MPa, 3/2000 mkg=ρ , %5=ξ , 33.0=υ

No.4 Firm clay, h=7m, G=30MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ

Base rock, G=1800MPa, 3/2300 mkg=ρ , %5=ξ , 33.0=υ


α α α


5-13

Figure 5-2. Amplification spectra of site 3, (a) horizontal out-of-plane motion;

(b) horizontal in-plane motion; and (c) vertical in-plane motion

Table 5-1. First two vibration frequencies of the sites in the

horizontal and vertical directions

Site Horizontal (Hz) Vertical (Hz)

2.55 5.25 Site 1

10.15 21.10

4.85 10.20 Site 2

14.60 30.70

1.05 2.20 Site 3

2.65 5.65

To illustrate the proposed algorithms in the paper, two numerical examples are chosen to

simulate spatially varying ground motion time histories at the three locations of the canyon

site shown in Figure 5-1. In the first example, the site amplification effect is included, the

ground motion time histories are simulated to be compatible with the power spectral

density functions modelled by Equation (5-5), and a coherency loss function. In the second

example, the spatial ground motion time histories are simulated to be compatible with the

multiple response spectra associated with the respective site conditions.

5.4.2 Example 1-PSD compatible ground motion simulation

In this example, the ground motion time histories at different locations of the ground

surface shown in Figure 5-1 are generated. The motion on the base rock is assumed to have

the same intensity and frequency contents and is modelled by the filtered Tajimi-Kanai

power spectral density function as

Γ+−

+

+−== 222222

222

2222

4

0 4)(41

)2()()()()(

ωωξωωωωξ

ωξωωωωωωω

ggg

gg

fffPg SHS (5-17)


5-14

where )(ωPH is a high pass filter function [23], which is applied to filter out energy at

zero and very low frequencies to correct the singularity in ground velocity and

displacement power spectral density functions. )(0 ωS is the Tajimi-Kanai power spectral

density function [1], gω and gξ are the central frequency and damping ratio of the Tajimi-

Kanai power spectral density function, ωf and ξf are the central frequency and damping

ratio of the high pass filter. In the analysis, the horizontal out-of-plane motion is assumed

to consist of SH wave only, while the in-plane horizontal and vertical motion are assumed

to be combined P and SV wave. The parameters of the horizontal motion are assumed as

sradg /10πω = , 6.0=gξ , πω 5.0=f , 6.0=fξ and 32 /0034.0 sm=Γ . These parameters

correspond to a ground motion time history with duration T=20s and peak ground

acceleration (PGA) g2.0 and peak ground displacement (PGD) 0.082m based on the

standard random vibration method [24]. The vertical motion on the base rock is also

modelled with the same filtered Tajimi-Kanai power spectral density function, but the

amplitude is assumed to be 2/3 of the horizontal component. It should be noted that if a

specific site and an earthquake scenario is considered, a stochastic ground motion

attenuation model can be easily used to replace the filtered Tajimi-Kanai power spectral

density function to represent the specific base rock motion.

The Sobczyk model [21] is selected to describe the coherency loss between the ground

motions at points 'i and 'j ( ji ≠ ) at the base rock:

)/cosexp()/exp()/cosexp()()( ''''''''''2

appjiappjiappjijiji vdivdvdiii αωβωαωωγωγ −⋅−=−= (5-18)

in which, )('' ωγ iji

is the lagged coherency loss, β is a coefficient which reflects the level of

coherency loss, 0005.0=β is used in the present paper, which represents highly correlated

motions; '' jid is the distance between the points 'i and 'j , and mdd 100'''' 3221

== is

assumed; α is the incident angle of the incoming wave to the site, and is assumed to be

60°; appv is the apparent wave velocity at the base rock, which is 1768m/s according to the

base rock property and the specified incident angle.


5-15

To model the temporal variation of the simulated ground motions, the simulated stationary

time histories are multiplied by the Jennings envelope function [25], which has the

following form

⎪⎩

⎪⎨

⎧

≤<−−≤<≤≤

=Tttttttttttt

t

nn

n

)](155.0exp[1

0)/()( 0

02

0

ζ (5-19)

with st 20 = and stn 10= in this study.

In the simulation, the sampling frequency and the upper cut-off frequency are set to be 100

Hz and HzN 25=ω , and the time duration is assumed to be T=20s. To improve the

computational efficiency, the ground motions are generated in the frequency domain by

using the FFT technique as discussed above, and N=2048 is used in the paper.

The three generated horizontal base rock motions are shown in Figure 5-3(a) and 5-3(b) for

acceleration and displacement respectively. The PGAs and PGDs of the simulated motions

are 2.27, 2.21, 2.33m/s2 and 0.0861, 0.0857, 0.0839m respectively, which are close to the

theoretical PGA of g2.0 and PGD of 0.082 m. Figure 5-4 shows the comparisons of the

power spectral densities of the generated time histories with the target filtered Tajimi-Kanai

spectral density function. It shows that power spectral densities of the simulated motions

match well with the target spectrum. Figure 5-5 shows the coherency loss functions

between the generated time histories and the Sobczyk model, good match can also be

observed except for '3'1γ in the high frequency range. This, however, is expected because

as the distance increases, the cross correlation between the spatial motions or their

coherency values decrease rapidly with the frequency. Previous studies (e.g., [26]]) revealed

that the coherency value of about 0.3 to 0.4 is the threshold of cross correlation between

two time histories because numerical calculations of coherency function between any two

white noise series result in a value of about 0.3 to 0.4. Therefore the calculated coherency

loss between two simulated time histories remains at about 0.4 even the model coherency

function decreases below this threshold value.


5-16

Figure 5-3. Generated base rock motions in the horizontal directions

(a) acceleration; and (b) displacement

Figure 5-4. Comparison of power spectral density of the generated base rock acceleration

with model power spectral density


5-17

Figure 5-5. Comparison of coherency loss between the generated base rock accelerations

with model coherency loss function

Assuming the incoming motions at the base rock consist of SH wave with an incident

angle of o60=α , the horizontal out-of-plane acceleration and displacement time histories

on the ground surface are shown in Figure 5-6. It is obvious that the site amplification

effects alter the frequency contents and increase the amplitudes of the incoming wave.

Different wave paths result in different site amplification effect. For the given example, the

PGAs and PGDs on the ground surface reach 4.32, 6.53, 3.31m/s2 and 0.0565, 0.0540,

0.0712m at the three different locations as shown in Figure 5-6. As compared with the

motions at the base rock, site 2 significantly amplifies the horizontal out-of-plane ground

acceleration. This is because the fundamental vibration frequency of site 2 is 4.85Hz as

given in Table 5-1, which is very close to the central frequency of the filtered Tajimi-Kanai

power spectral density function of ground motions at the base rock, so resonance occurs.

Site 1 and 3 also amplify the base rock motion, but with a less extent. It is interesting to

find that, though the site amplifies the PGA of surface motions, it is not necessarily result

in larger PGD. For the given example, the PGDs even decrease 34%, 37% and 15%

respectively. This might attribute to the fact that local soil layers also filter the frequency

content of the incoming waves. Although PGA of site 2 is the largest, the PGD is the

smallest because site 2 is the stiffest among the three sites. On contrary, PGA of site 3 is

the smallest, but PGD is the largest because site 3 is the softest. In general, the softer is the

site, the larger is the PGD. Figure 5-7 shows the comparisons of the simulated power

spectral densities with the theoretical values, good agreements are observed.


5-18

Figure 5-6. Generated horizontal out-of-plane motions on ground surface


For the coherency loss function between surface motions at a canyon site, the analysis

based on the recorded seismic data showed larger variability than that on the flat-lying sites

[27, 28]. Further studies revealed that local site effect not only causes phase difference of

the coherency function [10], but also affects its modulus [29, 30]. Figure 5-8 shows the

comparison of the lagged coherency loss functions of the base rock motions (solid line)

with those of the simulated surface motions (dashed line) in Figure 5-6. As shown, the

coherency loss between the surface motions is smaller than the corresponding base rock

motion. These results are consistent with those obtained from recorded surface ground

motions [27, 28]. Same results of coherency loss between in-plane surface ground motions,

which are not shown here, are also obtained. They indicate wave propagation through non-

uniform paths cause further coherency loss between spatial ground motions. More detailed


5-19

discussions of the influences of local site conditions on coherency loss of surface ground

motions can be found in Reference [30].

Figure 5-7. Comparison of power spectral density of the generated horizontal out-of-plane

acceleration on ground surface with the respective theoretical power spectral density

Figure 5-8. Comparison of the coherency loss functions between base rock motions

(solid line) and those between surface motions (dashed line)

Assuming the incoming motions on the base rock are combined P and SV waves with the

P wave incident angle o60 and SV wave incident angle o4.75 , the horizontal and vertical in-

plane motions on the ground surface are generated. Figure 5-9 shows the generated

horizontal in-plane acceleration and displacement time histories. The comparisons between

the theoretically derived power spectral densities and those of the generated time histories

are shown in Figure 5-10. As shown, the simulated time histories match the target spectral

density functions well. It can also be noticed that the horizontal in-plane motions on the

ground surface are similar to the simulated out-of-plane motions based on SH wave

assumption. This is expected because both the SH and SV waves have the same

characteristics as mentioned above. Figure 5-11 shows the simulated vertical in-plane

ground motion time histories. The comparisons of the spectral density functions of the

simulated time histories with the corresponding target functions are shown in Figure 5-12.

Good agreements are observed again. As expected, the simulated vertical motions are


5-20

different from the horizontal motions because the vertical motions have rather different

spectral density functions from the horizontal motion owing to the different vertical

vibration modes from the horizontal vibration modes of the site. As shown in Figure 5-11,

the PGAs and PGDs of the vertical motions are 3.11, 2.78, 2.67m/s2 and 0.0293, 0.0289,

0.0349m respectively for the three sites. Site 1 amplifies the base rock motion most because

the fundamental vertical vibration frequency of site 1 is 5.25Hz as given in Table 5-1,

which is close to the central frequency of the filtered Tajimi-Kanai power spectral density

function of ground motion at the base rock. Resonance results in the significant site

amplification.

Figure 5-9. Generated horizontal in-plane motions on ground surface



5-21

Figure 5-10. Comparison of power spectral density of the generated horizontal in-plane


Figure 5-11. Generated vertical in-plane motions on ground surface



5-22

Figure 5-12. Comparison of power spectral density of the generated vertical in-plane


The above results also indicate that surface ground motion energy may concentrate at one

or more frequency bands depending on the site vibration modes and the frequency of

incident motions to the site. Multi-layered site amplifies seismic wave energy at frequencies

around its vibration modes. Wave propagation in the soil site also results in a loss of spatial

ground motion coherency. Therefore it is important to model the wave propagation in the

local site to reliably predict surface ground motions.

5.4.3 Example 2 -Response spectrum compatible ground motion simulation

In this example, spatially correlated time histories on ground surface are generated to be

compatible with the design spectra for different site conditions specified in the New

Zealand Earthquake Loading Code [31]. The sub-soil classes at the three sites are assumed

to be shallow soil (Class C), rock (Class B) and deep/soft soil (Class D), respectively. The

peak ground acceleration (PGA) for the three sites is assumed to be 2m/s2. The

corresponding response spectra normalized to PGA of 2m/s2 in the New Zealand

Earthquake Loading Code for the three sites are plotted in Figure 5-14 (dashed line).

The Sobczyk model [21] is again selected to describe the coherency loss between the

ground motions at any two locations i and j. The seismic wave apparent velocity is assumed

as 1000m/s. It should be noted that the Sobczyk model for spatial ground motion

coherency is suitable for a flat site. As observed above and in a few previous studies [29,

30], this model overestimates coherency of spatial ground motions on surface of a canyon

site. However, it is adopted here in this example to model spatial ground motion coherency

loss because there is no suitable model available. Moreover, the coherency loss between

spatial ground motions on a canyon site is not well understood yet. Some previous studies

(e.g., [10]) also adopted the coherency model for ground motions on flat-laying site to


5-23

model spatial ground motions on non-uniform site. If a proper coherency model was

available, it could be easily implemented in the simulation procedure described above.

Figure 5-13. Generated time histories according to the specified design response spectra


The shape function in the form of Equation (5-19) is applied to modulate the simulated

stationary time histories. Figure 5-13 shows the generated acceleration and displacement

time histories at the three locations on the ground surface after 4 iterations with the

damping ratio 05.0=ξ and probability coefficient 85.0=p . The sampling frequency and

the upper cut-off frequency are set to be 100 and 25Hz, respectively. The time duration is

T=20s. As shown in Figure 5-13, though the PGAs for the three sub-soils are almost the

same, about 2m/s2, as defined in the design spectra, the PGDs are very different because

of the different frequency contents of ground motions corresponding to the different soil


5-24

conditions. The PGD of surface motion increases with the decrease in soil stiffness. In this

example, the PGD of the three sites reaches 0.0805, 0.0624 and 0.124m respectively. Figure

5-14 and Figure 5-15 show the response spectra and coherency loss function of the

generated time histories and the prescribed models, good matches are observed.

Figure 5-14. Comparison of the generated acceleration and the target response spectra

Figure 5-15. Comparison of coherency loss between the generated time histories with the

model coherency loss function

5.5 Conclusions

This paper presents a method to model and simulate spatially varying earthquake ground

motion time histories at sites with non-uniform conditions. It takes into consideration of

the local site effects on ground motion amplification and spatial variation. The base rock

motions can be modelled by using a filtered Tajimi-Kanai power spectral density function

or a stochastic ground motion attenuation model. The site specific ground motion power

spectral density function is derived by considering seismic wave propagations through the

local site by assuming the base rock motions consisting of out-of-plane SH wave and in-

plane combined P and SV waves with an incident angle to the site. The spectral

representation method is used to simulate the spatially varying earthquake ground motions.

It is proven that the simulated spatial ground motion time histories are compatible with the

respective target power spectral densities or design response spectra individually, and the


5-25

model coherency loss function between any two of them. This method can be used to

simulate spatial ground motions on a non-uniform site with explicit consideration of the

influences of the specific site conditions. It leads to a more realistic modelling of spatial

ground motions on non-uniform sites as compared to the common assumption of uniform

ground motion intensity in most previous studies. The simulated time histories can be used

as inputs to multiple supports of long-span structures on non-uniform sites in engineering

practice.

5.6 References


structure during an earthquake. Proc. of 2nd World Conference on Earthquake Engineering,

Tokyo, Japan, 1960; 781-796.

2. Clough RW, Penzien J. Dynamics of Structures. New York: McGraw Hill; 1993.

3. Joyner WB, Boore DM. Measurement, characterization and prediction of strong

ground motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in

Ground Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.

4. Atkinson GM, Boore DM. Evaluation of models for earthquake source spectra in

Eastern North America. Bulletin of the Seismological Society of America 1998; 88(4): 917-

934.

5. Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use

in Perth Western Australia. Soil Dynamics and Earthquake Engineering 2009; 29(5): 909-

924.

6. Wolf JP. Dynamic Soil-structure Interaction, Englewood Cliffs, NJ: Prentice Hall; 1985.

7. Wolf JP. Soil-structure interaction analysis in time domain, Englewood Cliffs, NJ: Prentice

Hall; 1988.

8. Safak E. Discrete-time analysis of seismic site amplification. Journal of Engineering

Mechanics 1995; 121(7): 801-809.

9. Zerva A, Zervas V. Spatial variation of seismic ground motions: An overview.

Applied Mechanics Reviews 2002; 56(3): 271-297.




simulation based on SMART-1 Array data. Nuclear Engineering and Design 1989; 111:

293-310.


5-26

12. Deodatis G. Non-stationary stochastic vector processes: seismic ground motion

applications. Probabilistic Engineering Mechanics 1996; 11(3): 149-167.

13. Shinozuka M. Monte Carlo solution of structural dynamics. Computers and Structures

1972; 2: 855-874.

14. Shinozuka M, Jan CM. Digital simulation of random processes and its applications.

Journal of Sound and Vibration 1972; 25(1): 111-128.

15. Shinozuka M, Deodatis G. Stochastic process models for earthquake ground

motion. Probabilistic Engineering Mechanics 1988; 3(3): 114-123.

16. Hao H. Input seismic motions for use in the seismic structural response analysis.

The Sixth International Conference on Soil Dynamics and Earthquake Engineering, 1993; 87-

100.

17. Wang S, Hao H. Effects of random vibrations of soil properties on site

amplification of seismic ground motions. Soil Dynamics and Earthquake Engineering

2002; 22(7): 551-564.

18. Idriss IM, Sun JI. User’s manual for SHAKE91, in User’s Manual for SHAKE91,

Department of Civil and Environmental Engineering, University of California at

Davis, 1992.

19. Baedet JP, Ichii K, Lin CH. EERA, a computer program for equivalent linear

earthquake site response analysis of layered soil deposits, in EERA, A Computer

Program for Equivalent Linear Earthquake Site Response Analysis of Layered Soil Deposits,

University of Southern California, 2000.

20. Baedet JP, Tobita T. NEAR, a computer program for nonlinear earthquake site

response analysis of layered soil deposits, in NEAR, A Computer Program for

Equivalent Linear Earthquake Site Response Analysis of Layered Soil Deposit, University of

Southern California, 2001.

21. Sobczky K. Stochastic Wave Propagation, Netherlands: Kluwer Academic Publishers;

1991.

22. Kaul MK. Stochastic characterization of earthquake through their response

spectrum,” Earthquake Engineering and Structural Dynamics 1978; 6:187-196.


earthquakes. Report No. UCB/EERC-69-03, University of California, Berkeley,

1969.


Mechanics 1980; 106: 1195-1213.


5-27

25. Jennings PC, Housner GW, Tsai NC. Simulated earthquake motions. Report of

Earthquake Engineering Research Laboratory, EERL-02, California Institute of

Technology, 1968.

26. Hao H. Effects of spatial variation of ground motions on large multiply-supported

structures. Report No. UCB/EERC-89-06, University of California, Berkeley, 1989.



10(1): 1-13.



Publication 2007; pp. 1-10.

29. Lou L, Zerva A. Effects of spatially variable ground motions on the seismic

response of s skewed, multi-span, RC highway bridge. Soil Dynamics and Earthquake

Engineering 2005; 25: 729-740.

30. Bi K, Hao H. Influences of irregular topography and random soil properties on


Structural Dynamics 2010, published online.

31. Standards New Zealand. Structural design actions, Part 5: Earthquake actions in

New Zealand (NZS1170.5-2004), 2004.


6-1

Chapter 6 Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions

By: Kaiming Bi and Hong Hao

Abstract: Coherency functions are used to describe the spatial variation of seismic ground

motions at multiple supports of long span structures. Many coherency function models

have been proposed based on theoretical derivation or measured spatial ground motion

time histories at dense seismographic arrays. Most of them are suitable for modelling

spatial ground motions on flat-lying alluvial sites. It has been found that these coherency

functions are not appropriate for modelling spatial variations of ground motions at sites

with irregular topography [1]. This paper investigates the influence of layered irregular sites

and random soil properties on coherency functions of spatial ground motions on ground

surface. Ground motion time histories at different locations on ground surface of the

irregular site are generated based on the combined spectral representation method and one-

dimensional wave propagation theory. Random soil properties, including shear modulus,

density and damping ratio of each layer are assumed to follow normal distributions, and are

modelled by the independent one-dimensional random fields in the vertical direction.

Monte-Carlo simulations are employed to model the effect of random variations of soil

properties on the simulated surface ground motion time histories. The coherency function

is estimated from the simulated ground motion time histories. Numerical examples are

presented to illustrate the proposed method. Numerical results show that coherency

function directly relates to the spectral ratio of two local sites, and the influence of

randomly varying soil properties at a canyon site on coherency functions of spatial surface

ground motions cannot be neglected.

Keywords: coherency loss function; irregular topography; random soil properties; Monte-

Carlo simulation


6-2

6.1 Introduction

For large dimensional structures, such as long-span bridges, pipelines, communication

transmission systems, their supports inevitably undergo different seismic motions during an

earthquake owing to the ground motion spatial variation. Past investigations indicate that

the effect of the spatial variation of seismic motions on the structural responses cannot be

neglected, and can be, in cases, detrimental [2]. Ground motion spatial variation effect has

been extensively studied by many researchers especially after the installation of strong

motion arrays (e.g. the SMART-1 array in Lotung, Taiwan). Many empirical [3-7] and semi-

empirical [8-9] models have been proposed mostly for flat-lying alluvial sites. These

coherency functions usually consist of two parts, the modulus or called lagged coherency,

which measures the similarity of the seismic motions between the two stations, and the

phase, which describes the wave passage effect, i.e., the delay in the arrival of the wave

forms at the further away station caused by the propagation of the seismic wave. It is

generally found that the lagged coherency decreases smoothly as a function of station

separation and wave frequency. To consider local site effect, Der Kiureghian [10] proposed

a theoretical model to describe coherency function of motions on the ground surface, in

which he assumed that site effect influences the phase of the coherency function only,

while it does not affect the lagged coherency.

Contrasting to the observations on the flat-lying sites, Somerville et al. [1] investigated the

coherency function of ground motions on a site located on folded sedimentary rocks (the

Coalinga anticline), and found that the lagged coherency does not show a strong

dependence on station separation and wave frequency, and the incoherency is generally

higher than that on the flat-lying sites. They attributed the chaotic behaviour to the wave

propagation in a medium having strong lateral heterogeneities in seismic velocity. Liao et al.

[11], based on the seismic data recorded at the Parkway array in Wainuiomata Valley, New

Zealand, compared the lagged coherency functions of different station combinations, i.e.,

four groups with station pairs located on the sediments, one group with one sedimentary

station and one rock station. They concluded that the lagged coherency between the

sediment and rock stations exhibit large variability and follow no consistent pattern. These

observations suggest that the spatial coherency function measured on flat-lying sedimentary

sites may not provide a good description of spatial ground motion coherencies on sites

with irregular topography. These observations also indicate that the theoretical model

proposed by Der Kiureghian [10] might not be able to reliably describe the influence of

local site effect on the coherency function.


6-3

Although it was observed that the heterogeneity of site conditions strongly affect the

ground motion spatial variations [1, 11], all the previous studies and theoretical and

empirical coherency models mentioned above assumed the site characteristics are fully

deterministic and homogeneous. However, in reality, there always exist spatial variations of

soil properties and uncertainties in defining the properties of soils. This results from the

natural heterogeneity or variability of soils, the limited availability of information about

internal conditions and sometimes the measurement errors. These uncertainties associated

with system parameters are also likely to have influence on the coherency function. Zerva

and Harada [12] modelled horizontal soil layers at a site as a 1-DOF system with random

characteristics to study the effect of uncertain soil properties on the coherency function.

They pointed out that the spatial coherency of motions on the ground surface is similar to

that of the incident motion at the base rock except at the predominant frequency of the

layer, where it decreases considerably. The effect of uncertain soil properties should also be

incorporated in spatial variation model of ground motions. Their explanation for this

phenomenon was that for input motion frequencies close to the mean natural frequency of

the ‘oscillators’, the response of the systems was affected by the variability in the value of

this natural frequency, and resulted in loss of correlation [12]. However, it should be noted

that a 1-DOF system cannot realistically represent the multiple predominate frequencies

that may exist at a site with multiple layers and multiple modes. Liao and Li [13] developed

an analytical stochastic method to evaluate the seismic coherency function, in which a

numerical approach to compute coherency function is developed by combining the

pseudo-excitation method with wave motion finite element simulation techniques. An

orthogonal expansion method is introduced to study the effect of uncertain soil properties

on the coherency function. The results also demonstrate that the lagged coherency values

tend to decrease in the vicinity of the resonant frequencies of the site. This method is,

however, difficult to be implemented and sometimes a little arbitrary to select the

absorbing boundary conditions, and is difficult to explain why the lagged coherency

function varies significantly over relatively short distances owing to the inherent limitations

of using finite element method to model wave motion in a unbounded medium [14].

It is obvious that the effects of irregular topography and random soil properties of a site on

the coherency function of spatial ground motions cannot be neglected. However, at the

present, only very limited recorded spatial ground motion data on sites of different

conditions are available. They are not sufficient to determine the general spatial

incoherence characteristics of ground motions and derive empirical relations to model

spatial ground motion variations at a site with varying site conditions. On the other hand,


6-4

to the best knowledge of the authors, no more theoretical/analytical analysis in this field

can be found except for the studies mentioned above [10, 12, 13].

The present study investigates the influence of a layered canyon site and randomly varying

soil properties on coherency function of spatial ground motions. The site is assumed

consisting of horizontally extended multiple soil layers on a half-space (base rock). The

base rock motions at different locations are assumed to have the same intensity, and are

modeled by a filtered Tajimi-Kanai power spectral density function. The spatial variation of

ground motions on the base rock is accounted for by an empirical coherency function for

spatial ground motions on a flat-lying site. Using the one-dimensional wave propagation

theory [15], the power spectral density functions of spatial ground motions at various

locations on surface of the canyon site can be derived by assuming the base rock motions

consisting of out-of-plane SH wave or in-plane combined P and SV waves propagating into

the site with an assumed incident angle. The spatially varying ground motion time histories

can then be generated based on the spectral representation method. In order to take into

consideration the random soil properties, Monte-Carlo simulation method is used in the

study. The random soil properties considered include the shear modulus, density and

damping ratio of each layer, and they are all assumed to have normal distributions in the

vertical direction and are modelled as independent one-dimensional random fields [16]. In

numerical calculations, for each realization of the random soil properties, spatial ground

motion time histories are generated. These time histories are then used to calculate the

lagged coherency between any two ground motion time histories. The numerical

calculations include the following steps: 1) random generation of soil properties; 2)

estimation of ground motion power spectral density functions at various points on the

canyon surface; 3) simulations of spatial ground motion time histories; and 4) calculations

of coherency functions. These steps are repeated until the estimated mean and standard

deviation of the lagged coherency between ground motions at any two points converge.

Numerical examples are presented to demonstrate the proposed method and to study the

effects of irregular topography and random soil properties on coherency function of spatial

ground motions.


6-5

6.2 Theoretical basis

6.2.1 Estimation of coherency function

Let )(tu j and )(tuk be the recorded (simulated) acceleration time histories at locations j

and k of a site, and the corresponding Fourier transform of the time histories are )(ωjU

and )(ωkU , respectively. The smoothed auto spectral density function of ground motion at

location j or k is then

( ) ( ) ( ) kjimUmWSM

Mmninii or 2 =Δ+Δ= ∑

−=

ωωωω (6-1)

and the cross power spectral density function between motions at stations j and k is

( ) ( ) ( ) ( )∑−=

∗ Δ+Δ+Δ=M

Mmnknjnjk mUmUmWS ωωωωωω (6-2)

where ( )ωW is the spectral smoothing window, ωΔ is the frequency step, ωω Δ= nn is

the n-th discrete frequency, and ∗ denotes the complex conjugate.

The coherency function of the spatial ground motions can be obtained as [4]

)()()(

)(ωω

ωωγ

kkjj

jkjk SS

S= (6-3)

The coherency function in Equation (6-3) is generally a complex function and can be

written as

[ ])(exp)()( ωθωγωγ jkjkjk i= (6-4)

in which )(ωγ jk is the lagged coherency, ( )( )( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡= −

ωω

ωθjk

jkjk S

SReIm

tan)( 1 is the phase angle,

‘Im’ and ‘Re’ denote the imaginary and real part of a complex number.


6-6

Based on the analysis above, the coherency function can be readily estimated if the

acceleration time histories at each location are available. The simulation of ground motion

time histories is based on the one-dimensional wave propagation theory [17] and the

spectral representation method. These two parts are briefly introduced in Sections 6.2.2

and 6.2.3, more detailed information can be found in Reference [15].

6.2.2 One-dimensional wave propagation theory

For a site with horizontally extended multiple soil layers on a half space (base rock), the

base rock motions can be assumed to consist of out-of-plane SH wave or in-plane

combined P and SV waves propagating into a site with an assumed incident angle. For a

harmonic excitation with frequencyω , the dynamic equilibrium equations can be written as

[17]

ec

ep2

22 ω

−=∇ or { } { }Ω−=Ω∇ 2

22

scω (6-5)

where e2∇ and { }Ω∇2 are the Laplace operator of the volumetric strain amplitude e and

rotational-strain-vector { }Ω . pc and sc are the P- and S-wave velocity, respectively. This

equation can be solved by using the P- and S-wave trial function. The out-of-plane

displacements with the amplitude v is caused by the incident SH wave, while the in-plane

displacements with the amplitude u and w in the horizontal and vertical directions depend

on the combined P and SV waves. The amplitude v is independent of u and w, hence, the

two-dimensional dynamic stiffness matrix of each soil layer for the out-of-plane and in-

plane motion, ][ LSHS and ][ L

SVPS − , can be formulated independently by analysing the

relations of shear stresses and displacements at the boundary of each soil layer. Assembling

the matrices of each soil layer and the base rock, the dynamic stiffness of the total system is

obtained and denoted by ][ SHS and ][ SVPS − , respectively. The dynamic equilibrium

equation of the site in the frequency domain is thus [17]

{ } { }SHSHSH PuS =][ or { } { }SVPSVPSVP PuS −−− =][ (6-6)

where { }SHu and { }SHP are the out-of-plane displacements and load vector corresponding

to the incident SH wave, { }SVPu − and { }SVPP − are the in-plane displacements and load

vector of the combined P and SV waves. The stiffness matrices ][ SHS and ][ SVPS − depend


6-7

on soil properties, incident wave type, incident angle and circular frequency ω . The

dynamic load { }SHP and { }SVPP − depend on the base rock properties, incident wave type,

incident wave frequency and amplitude. By solving Equation (6-6) in the frequency domain

at every discrete frequency, the relationship of the amplitudes between the base rock and

each soil layer can be formed, and the site transfer function )]([ ωH in the out-of-plane and

in-plane directions can be estimated.

6.2.3 Ground motion generation

Consider a canyon site with horizontally extended multiple soil layers resting on an elastic

half-space as shown in Figure 6-1, in which hm, Gm, mρ , mξ and mυ is the depth, shear

modulus, mass density, damping ratio and Poisson’s ratio of layer m. The spatially varying

base rock motions are assumed to consist of out-of-plane SH wave or in-plane combined P

and SV waves and propagating into the layered soil site with an assumed incident angle as

discussed above. The incident motions at different locations on the base rock are assumed

to have the same power spectral density, and are modelled by a filtered Tajimi-Kanai [18]

power spectral density function. The spatial variation of ground motions at base rock is

modelled by an empirical coherency function for spatial ground motions on a flat site. The

cross power spectral density functions of surface motions at n locations of the layered site

can be written as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅

=

)()()(

)()()()()()(

)(

21

22221

11211

ωωω

ωωωωωω

ω

nnnn

n

n

SiSiS

iSSiSiSiSS

iS (6-7)

where

nkjidSiHiHiS

njSiHS

kjkjgkjjk

gjjj

,...,2,1,),()()()()(

,...,2,1)()()(

''''*

2

==

==

ωγωωωω

ωωω (6-8)

are the auto and cross power spectral density function respectively. In which )(ωgS is the

ground motion power spectral density on the base rock; ),( '''' ωγ idkjkj

is the coherency

function between location 'j and 'k on the base rock; )( ωiH j , )( ωiH k are the site transfer


6-8

function at locations j and k on the ground surface, which can be formulated based on one-

dimensional wave propagation theory discussed in Section 6.2.2.

Figure 6-1. Schematic view of a layered canyon site

Decomposing the Hermitian, positive definite matrix )( ωiS into the multiplication of a

complex lower triangular matrix )( ωiL and its Hermitian )( ωiLH

)()()( ωωω iLiLiS H= (6-9)

the stationary time series njtu j ,...,2,1),( = , can be simulated in the time domain directly

[6]

)]()(cos[)()(1 1

nmnnjmnn

j

m

N

njmj tAtu ωϕωβωω ++= ∑∑

= =

(6-10)

where

Njm

jmjm

Njmjm

iLiL

iLA

ωωωω

ωβ

ωωωωω

≤≤=

≤≤Δ=

− 0),)](Re[)](Im[

(tan)(

0,)(4)(

1

(6-11)

are the amplitudes and phase angles of the simulated time histories which ensure the

spectra of the simulated time histories compatible with those given in Equation (8);

)( nmn ωϕ is the random phase angles uniformly distributed over the range of ]2,0[ π , mnϕ

and rsϕ should be statistically independent unless rm = and sn = ; Nω represents an

upper cut-off frequency beyond which the elements of the cross power spectral density

matrix given in Equation (6-7) is assumed to be zero.

k

j

j’ k’

Layer l: ,lh lG , lρ , lξ , lυ

M

M

Layer m: ,mh mG , mρ , mξ , mυ

Layer m-1: ,1−mh 1−mG , 1−mρ , 1−mξ , 1−mυ

Layer 1: ,1h 1G , 1ρ , 1ξ , 1υ

Base rock: BG , Bρ , Bξ , Bυ


6-9

The generated time series by Equation (6-10) are stationary processes. In order to obtain

the non-stationary time histories, an envelope function )(tζ is applied to )(tu j . The non-

stationary time histories at different locations are then obtained by

njtuttf jj ,...,2,1),()()( ==ζ (6-12)

6.2.4 Random field theory

In engineering practice there are always some uncertainties in the soil properties because of

the reasons mentioned above. The random field theory [16] is widely used to describe the

variability of soil properties. In this theory the random soil property )(zu is characterized

by the mean value u , standard deviation uσ and the correlation distance uδ . uσ measures

the intensity of fluctuation or degree to which actual value of )(zu may deviate from. uδ

measures the correlation level or persistence of the property from one point to another in a

site, small values of uδ suggest rapid fluctuation about the average, while large values of uδ

imply a slowly varying component is superimposed on the average value of u .

Consider a one-dimensional random field )(zu with mean value )(zu and standard

deviation uσ , its local average process )(zuZ of )(zu over the interval Z centered at z is

defined as:

')'(1)(2/

2/dzzu

Zzu

Zz

ZzZ ∫+

−= (6-13)

It can be seen that the local average )(zuZ depends on the specific location of the interval

z within the statistically homogeneous soil layer. The mean and variance of )(zuZ are [16]

[ ] [ ]

[ ] )()(

)()()(

2 ZzuVar

zuzuEzuE

uZ

Z

λσ=

== (6-14)

where )(Zλ is a variance reduction function of )(zu , which measures the reduction of

point variance 2uσ under local average. The variance function )(Zλ can be derived from

auto-correlation function )( zu Δρ in the following form


6-10

)()()1(2)(0

zdzZz

ZZ u

ZΔΔ

Δ−= ∫ ρλ (6-15)

By using the exponential auto-correlation function [19]

)/2exp()( uu zz δρ Δ−=Δ (6-16)

the variance reduction function can be derived as [19]

[ ]1)/(2)/(2

1)( )/(22 −+= − uZ

uu

eZZ

Z δδδ

λ (6-17)

In this study, the shear modulus, density and damping ratio of each soil layer of the site are

regarded as random fields, and are assumed to follow normal distributions in the vertical

direction. These random fields can be modelled by introducing the mean value, standard

deviation and correlation distance of each parameter as mentioned above. Take shear

modulus as an example

( )φλφλσ )(1)( ZCOVGZGG G ×+=+= (6-18)

where G and Gσ are the mean value and standard deviation of shear modulus, )(Zλ is

the variance reduction function and φ is a normal distributed random process with zero

mean and unity variance. GCOV G /σ= is the coefficient of variation.

6.2.5 Monte-Carlo simulation

Monte-Carlo simulations have been extensively used in many scientific fields with random

parameters. It was found that for the range of variability usually present in soil properties,

Monte-Carlo based method, though computationally intensive, might be the simplest and

most direct method. Other methods, which are basically expansion based, do not provide

accurate results when the coefficients of variation of soil properties are large [20]. In this

study, Monte-Carlo simulations are also employed to account for the influence of random

soil properties on spatial ground motions. In Monte-Carlo simulations, soil properties are

randomly generated according to their distributions. Each set of random soil properties are

considered as deterministic in estimating the power spectral densities of ground motions.


6-11

Then spatial ground motion time histories are simulated according to the procedures

described above.


To study the influence of irregular topography and random soil properties on the

coherency function between different motions on the ground surface, a four-layer canyon

site resting on the base rock is selected as an example as shown in Figure 6-2. The mean

values of the corresponding soil properties of each soil layer and base rock are also given in

the Figure.

The motions on the base rock are assumed to have the same intensities and frequency

contents and are modelled by the filtered Tajimi-Kanai power spectral density function in

the following form:

Γ+−

+

+−== 222222

222

2222

4

0 4)(41

)2()()()()(

ωωξωωωωξ

ωξωωωωωωω

ggg

gg

fffPg SHS (6-19)

where )(ωPH is a high pass filter function [21], which is applied to filter out energy at zero

and very low frequencies to correct the singularity in ground velocity and displacement

power spectral density functions. )(0 ωS is the Tajimi-Kanai power spectral density

function [18], gω and gξ are the central frequency and damping ratio of the Tajimi-Kanai

power spectral density function, ωf and ξf are the corresponding central frequency and

damping ratio of the high pass filter. Γ is a scaling factor depending on the ground motion

intensity. In the analysis, the out-of-plane horizontal motion is assumed to consist of SH

wave only, while the in-plane horizontal and vertical motions are assumed to be combined

P and SV waves. The parameters for the horizontal motion are assumed as πω 10=g rad/s,

6.0=gξ , πω 5.0=f , 6.0=fξ and 0034.0=Γ m2/s3. These parameters correspond to a

ground motion time history with duration 20=T s and peak ground acceleration (PGA)

0.2g based on the standard random vibration method [22]. The vertical motion on the base

rock is also modelled with the same filtered Tajimi-Kanai power spectral density function,

but the amplitude is assumed to be 2/3 of the horizontal component of PGA 0.2g.


6-12

Figure 6-2. A four-layer canyon site with deterministic soil properties (not to scale)


motions at points j’ and k’ on the base rock:


appkjappkjappkjkjkj vdivdvdiii αωβωαωωγωγ −⋅−=−= (6-20)

where β is a coefficient reflecting the level of coherency loss, 001.0=β is used in the

present paper, which represents intermediately correlated motions; ''kjd is the distance

between the points j’ and k’, and =''kjd 100 m is assumed; α is the incident angle of the

incoming wave to the site, and is assumed to be 60°; appv is the apparent wave velocity on

the base rock, which is 1768 m/s according to the base rock property and the specified

incident angle. Seismic waves are assumed propagating vertically from the base rock to the

ground surface.

Take the canyon site with deterministic soil properties as an example. Assuming the soil

properties of each soil layer equal to their mean values as given in Figure 6-2, the

acceleration time histories on the base rock and the ground surface are simulated based on

the procedures presented in Sections 6.2.2 and 6.2.3. The sampling frequency and the

upper cut-off frequency are set to be 100 Hz and 20=Nω Hz, respectively. 2048 sampling

points are used in each set of ground motion time histories. As mnϕ in Equation (6-10) is a

random variable uniformly distributed over the range of ]2,0[ π , any realization of a

random angle mnϕ will result in a generation of a set of spatial ground acceleration time

histories which are compatible with the spectral density function in Equation (6-8). Figure

6-3 shows one set of the simulated acceleration time histories.

No.3 Soft Clay, h=15m, G=20MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ


No.1 Firm clay, h=12m, G=30MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ

Base rock: G=1800MPa, 3/2300 mkg=ρ , %5=ξ , 33.0=υ

j’ k’

j

No.4 Sandy fill, h=5m, G=30MPa, 3/1900 mkg=ρ , %5=ξ , 40.0=υ

k


6-13

Figure 6-3. Simulated acceleration time histories

The coherency function between different motions on the ground surface can be estimated

after the generation of acceleration time histories. However, it needs to be emphasized that

coherency estimates depend strongly on the type of the smoothing window and the

amount of smoothing performed on the raw data. Abrahamson et al. [24] noted that the

choice of the smoothing window should be directed not only from the statistic properties

of the ground motion time histories, but also from the problem for which it is analysing, so

that the required resolution is not lost. They suggested an 11-point Hamming window, if

the coherency estimates is to be used in structural analysis, for time windows less than

approximately 2000 samples and for structural damping coefficient 5% of critical [24]. It

should also be noted that if no smoothing is performed on the raw data, the lagged

coherency will always be unity for each frequency, and no information about the coherency

can be extracted from the data.

To obtain the mean lagged coherency functions on the base rock and ground surface,

Monte-Carlo simulation method is used as discussed in Section 6.2.5. Convergence test

needs to be conducted to check the number of Monte-Carlo simulations required to obtain

converged simulation results. Since a larger number of Monte-Carlo simulations is required

for the simulation to converge if the random variables under consideration have larger


6-14

COV, the case with the largest COV considered in this study, i.e., a COV of 60% for shear

modulus and damping ratio of each soil layer and 5% for soil density, which will be further

discussed in Section 6.3.2, is used to perform the convergence test. The mean values and

standard deviations of the lagged coherency function of the horizontal out-of-plane motion

at 0.2Hz, 2.0Hz, 5.0Hz and 9.0Hz are used as the quantity for convergence test. As shown

in Figure 6-4, the corresponding values virtually unchanged after 600 simulations,

indicating the simulations converged with 600 simulations. Results of the simulated in-

plane motions, which are not shown, also converge after 600 simulations. Therefore, 600

simulations are performed for each case in the subsequent calculations. Figure 6-5 shows

the comparison between the mean lagged coherency functions from the 600 simulated

spatial ground motion time histories on the base rock smoothed by the 11-point Hamming

window with the target model. It is evident that very good agreement can be obtained

except for the frequencies near zero. In fact, theoretically, coherency should tend to be

unity as frequency tends to zero, however, coherency estimates from ground motion time

histories, due to smoothing, can rarely reach this value.

Figure 6-4. Mean values and standard deviations of the lagged coherency of the horizontal

out-of-plane motion at 0.2, 2.0, 5.0 and 9.0Hz

Figure 6-5. Comparison of the mean lagged coherency on the base rock from 600

simulations with the target model


6-15

6.3.1 Influence of irregular topography

Assuming all the soil properties are deterministic and equals to their mean values, the

influence of irregular topography is studied first. Figure 6-6 shows the mean values of the

lagged coherency functions between the spatial ground motions of points j and k on the

ground surface of the canyon site. For comparison purpose, the lagged coherency between

incident motion on the base rock at j’ and k’ is also plotted. Figure 6-7 shows the

corresponding standard deviations. As shown, the standard deviations have a general trend

of increasing with frequency, but are relatively small, all less than 0.13. This indicates that

the lagged coherency is more difficult to be accurately modelled at high frequencies.

Nonetheless, as the standard deviations are relatively small as compared to the mean lagged

coherency values, including them will change the lagged coherency value, but not the

overall trend. Figure 6-6 shows that the coherency function between surface ground

motions differs from that between base rock motions significantly. At all frequencies, the

coherency loss functions on the ground surface are smaller than those on the base rock,

i.e., the coherency function on the base rock is the upper bound of the coherency of spatial

ground motions on the surface of a canyon site. This conclusion is in agreement with that

of Lou and Zerva [25], and Liao et al. [11]. It indicates that wave propagation through a

local site even with deterministic site properties further reduces the cross correlation

between spatial ground motions on the base rock. As shown, there are many obvious peaks

and troughs in the coherency function of surface motions. These peaks and troughs

directly relate to the modulus of the spectral ratio of two local sites, namely

( ) ( )ωω iHiH jk / , as shown in Figure 6-8. ( )ωiH j and ( )ωiH k are the transfer functions of

site j and k respectively. They are the spectral ratio of the surface motion at j or k to the

corresponding bedrock motion at j’ or k’, which can be calculated based on the one-

dimensional wave propagation theory as discussed in Section 6.2.2. Figure 6-9 shows the

modulus of the transfer functions at sites j and k. It is obvious that site amplifies the

motions on the base rock significantly, which makes the energy of surface ground motions

concentrate at a few frequencies corresponding to the various vibration modes of the site.

This result indicates the importance of considering the multiple modes of a local soil site

when estimating the seismic wave propagation and site amplification. The present result is

an extension of those obtained with a 1-DOF model [12]. With a 1-DOF model, the

influence of the higher vibration modes of the site on site amplification and hence the

spatial ground motion coherency cannot be included. Comparing Figure 6-6 with Figure 6-

8, it can be noted that when the spectral ratios differ from each other, the spatial ground

motions on the ground surface are least correlated with a minimum lagged coherency value.


6-16

Taking the horizontal out-of-plane motion as an example, four obvious minima can be

observed around the frequencies 0.78, 1.90, 4.20 and 7.10 Hz, which correspond to the

four evident peaks in the spectral ratio as shown in Figure 6-8(a). Similar conclusions can

be obtained for the in-plane motions. This is expected because the lagged coherency

measures the similarity of the motions at two different locations. If two sites amplify the

ground motions to the same extent at certain frequencies, the coherency loss is mainly

caused by the incoherence effect and wave passage effect, local site effect has little

influence on the lagged coherency. However, if the site amplification spectra are different

from each other at certain frequencies, the local site effect on wave propagation is

different. Therefore surface ground motions will be different at these frequencies, which

results in spatial surface ground motions less correlated. These observations coincide with

the recorded data from the Coalinga anticline in California [1] and the Wainuiomata Valley

in New Zealand [11]. These observations also indicate that site effect will not only cause

phase difference of the coherency function [10], but will also affect its modulus.

Liao and Li [13] used the auto-power spectral density of ground motion at one location of

the site to identify the lagged coherency function on the ground surface, and concluded

that the surface layer irregularity of a site can reduce the lagged coherency function values

in the vicinity of the resonant frequencies of the site. To examine their observation, the

horizontal out-of-plane motion of site j is used as an example. The fundamental vibration

frequency of the site is about 1.25 Hz as shown in Figure 6-9(a). According to Liao and Li’s

conclusion, the lagged coherency should have a minimum value at this frequency.

However, the present results actually display a peak value in the lagged frequency at this

frequency as shown in Figure 6-6(a). This contradicts with Liao and Li’s conclusion. This is

because in the present example, both wave paths from j’ to j and k’ to k or both sites

amplify the bedrock motion around this frequency, although to a different extent.

Therefore wave propagation through the two sites does not significantly reduce the cross

correlation of spatial bedrock motions at this frequency. This observation demonstrates

that using the amplitude of the power spectral density of ground motion at just one

location to assess the influence of wave propagation in an canyon site and hence the

coherency function of spatial surface ground motions may not lead to a reliable coherency

estimation. The spectral ratio between the two considered sites or two wave paths is a more

reliable and appropriate parameter to measure the local site effect on cross correlation of

spatial surface ground motions.


6-17

Figure 6-6. Comparison of the mean lagged coherency between the surface motions (j, k)

with that of the incident motion on the base rock

Figure 6-7. Standard deviations of the lagged coherency on the ground surface

Figure 6-8. Modulus of the site amplification spectral ratio of two local sites

Figure 6-9. Amplitudes of the site amplification spectra of two local sites


6-18

6.3.2 Influence of random soil properties

The influence of randomly varying soil properties on the coherency loss functions between

the surface motions is studied in this section. Without losing generality, assuming shear

modulus, damping ratio and soil density are random fields in all soil layers, and all follow a

normal distribution. The mean values of soil properties in every layer are given in Figure 6-

2. According to a more specific review and summary [26], in most common field

measurements, the coefficients of variation (COV) for the cohesion and undrained strength

of clay and sand are in a range of 10% to 100%. The statistical variation of the soil density

is, however, relatively small as compared with other soil parameters. Therefore, in the

present study, it is assumed that the shear modulus and damping ratio have COV of 20%,

40% and 60% for all soil layers, while the COV of soil density is assumed to be 5% in all

the cases. Vanmarcke [16] studied the scale of soil fluctuation, and concluded that the

correlation distance of various soils vary from 0.16 to 46m. For typical clay, it is about 5 m.

The correlation distance of 4 m is used in the present paper. It should be noted that in the

present study, only the random fluctuations of soil properties in the vertical direction are

considered, those in the horizontal direction are neglected because seismic waves are

assumed propagating vertically and modeled with the one-dimensional wave propagation

theory.

Figure 6-10. Influence of uncertain soil properties on

the mean values of lagged coherency functions


6-19

Figure 6-11. Influence of uncertain soil properties on the

standard deviations of lagged coherency functions

Figure 6-12. Influence of uncertain soil properties on the

mean spectral ratios of two local sites


6-20

Figure 6-10 and Figure 6-11 show the influence of random variations of soil properties on

the mean values and standard deviations of the lagged coherencies of spatial surface

motions. For comparison purpose, the corresponding values with deterministic soil

properties (COV=0), and that of the incident motion on the base rock are also plotted. As

shown, the influence of random soil properties on the lagged coherency between the

motions on the ground surface should not be neglected, especially for in-plane motions.

The lagged coherency between the motions on the ground surface is smaller than the

incident motion on the base rock as observed above. When the COV of soil properties is

0.2, the mean lagged coherencies are similar to those obtained by deterministic analysis.

Increasing COV of soil properties in general leads to smaller lagged coherencies between

the motions on the ground surface, but could result in larger coherency values at certain

frequencies where the spectral ratios of the two sites differ from each other significantly as

shown in Figure 6-12. In this case, larger COV leads to smaller spectral ratios, which results

in the relatively larger lagged coherency values. As shown in Figure 6-11, larger COV of

soil properties results in larger variations of the lagged coherency function on the ground

surface, as expected. It should be noted that these observations are based on the simulated

data from a canyon site. If a flat site is under consideration, and the randomness of soil

properties in the horizontal direction is neglected, the two local sites amplify ground

motions on the base rock to the same extent although randomness in the vertical direction

is considered. In this case, the spectral ratios of two local sites equal unity, and the

coherency function on the ground surface is then the same as that on the base rock

(incident motion). The random soil properties have no influence on the coherency function

on the ground surface in this case. This observation proves again that the influences of

local site on surface ground motion spatial variations depend on the similarity of the two

wave paths. If the two wave paths are the same, local site will not affect the surface ground

motion spatial variations.

6.3.3 Influence of random variation of each soil parameter

To investigate the effect of random variation of each soil parameter on the lagged

coherency function between different motions on the ground surface, assuming only one

soil parameter, namely either shear modulus, soil density or damping ratio, is random, while

the other two parameters are assumed to be deterministic in the calculation. The COVs for

shear modulus and damping ratio are assumed to be 40% and the COV for soil density is

assumed to be 5%. Figure 6-13 and Figure 6-14 shows the mean values and the

corresponding standard deviations of the lagged coherency respectively. The corresponding


6-21

values with deterministic soil properties, and that between the incident motions on the base

rock are plotted again for comparison purpose. As shown, mean values and standard

deviations of the lagged coherency obtained by considering only the damping ratio or soil

density as random parameter are almost the same as those with deterministic soil property

assumption, indicating the influence of random damping ratio and soil density on lagged

coherency is insignificant and can be neglected. On the other hand, the influence of the

random variations of shear modulus is obvious especially for the horizontal motions. These

results can be explained by the spectral ratios of the two local sites as shown in Figure 6-15,

in which the influence of random damping ratio and soil density on the spectral ratios is

insignificant while the influence of the shear modulus is pronounced. Because the lagged

coherency function directly relates to the spectral ratios of two local sites as discussed

above, this leads to the observations of lagged coherency functions in Figure 6-13 and

Figure 6-14.

It should be noted that all the results obtained above are based on the assumption of a

correlation distance uδ of 4 m for a typical clay site. In fact, the correlation distance varies

in a relatively wide range [16], when larger correlation distance is considered, similar

conclusions can be obtained but more prominent variation will be observed. These results

are not shown in the current paper owing to the page limit.

Figure 6-13. Influence of each random soil property on the

mean values of lagged coherency functions


6-22


standard deviations of lagged coherency functions


mean spectral ratios of two local sites


6-23

6.4 Conclusions

This paper evaluates the influence of local site irregular topography and random soil

properties on the coherency function between spatial surface motions. Following

conclusions are drawn:

1. The coherency function between surface ground motions on a canyon site is

different from that between base rock motions. The lagged coherency function on

the base rock is the upper bound of that on the ground surface.

2. For a canyon site, the coherency function of spatial surface ground motions

oscillates with frequency. The maximum and minimum coherency values are related

to the spectral ratios of two local sites or two wave paths. When the spectral ratios

of two local sites differ from each other significantly, the spatial ground motions on

the ground surface are least correlated. The coherency function models for motions

on a flat-lying site cannot be used to model that of motions on a canyon site.

3. The influence of random soil properties on the lagged coherency function depends

on the level of variations of soil properties. In general, the more significant are the

random variations of soil properties, the larger is the local site effect on spatial

surface ground motion variations. The random variations of soil damping ratio and

density have insignificant effect on the lagged coherency as compared to the

random variations of shear modulus.

It should be noted that the soil nonlinearities also affect the surface motion spatial

variations, but are not considered in the present paper. It is suggested to monitor some

canyon sites to check the results obtained in the present paper. Further study is also needed

to develop analytical or empirical relation of local site characteristics with ground motion

spatial variations for easy use in engineering application.

6.5 References



10(1):1-13.

2. Saxena V, Deodatis G, Shinozuka M. Effect of spatial variation of earthquake

ground motion on the nonlinear dynamic response of highway bridges. Proceeding of

12th World Conference on Earthquake Engineering, Auckland, New Zealand, 2000.


6-24

3. Loh CH. Analysis of the spatial variation of seismic waves and ground movement

from SMART-1 data. Earthquake Engineering and Structural Dynamics 1985; 13(5): 561-

581.



5. Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential


596.



111(3):293-310.

7. Harichandran RS. Estimating the spatial variation of earthquake ground motion

from dense array recordings. Structural Safety 1991; 10: 219-233.

8. Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground

motion. Earthquake Engineering and Structural Dynamics 1986; 14(6): 891-908.

9. Somerville PG, McLaren JP, Saikia CK, Helmberger DV. Site-specific estimation of

spatial incoherence of strong ground motion. Earthquake Engineering and Structural

Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special

Publication No. 20, 1988; 188-202.





Publication No. 170, 2007; 1-10.

12. Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion

coherence and strain estimations. Soil Dynamics and Earthquake Engineering 1997; 16:

445-457.

13. Liao S, Li J. A stochastic approach to site-response component in seismic ground

motion coherency model. Soil Dynamics and Earthquake Engineering 2002; 22: 813-

820.

14. Chen Y, Li J. Effect of random media on coherency function of seismic ground

motion. World Earthquake Engineering 2007; 23(3):1-6 (in Chinese).

15. Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform

intensities and frequency content. Australia Earthquake Engineering Society 2008

Conference, Ballart, Australia, 2008; Paper No. 18.


6-25

16. Vanmarcke EH. Probabilistic modelling of soil profiles. Journal of the Geotechnical

Engineering Division 1977; 103(11): 1227-1246.

17. Wolf JP. Dynamic soil-structure interaction. Prentice Hall: Englewood Cliffs, NJ, 1985.




19. Vanmarcke EH. Random fields: analysis and synthesis. Cambridge: MIT Press, 1983.

20. Yeh CH, Rahman MS. Stochastic finite element methods for the seismic response

of soils. Internal Journal for Numerical and Analytical Methods in Geomechanics, 1998;

22(10): 819-850.


earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley;

1969.

22. Der Kiureghian A. Structural response to stationary excitation. Journal of the

Engineering Mechanics Division 1980; 106(6): 1195-1213.

23. Sobczky K. Stochastic wave propagation. Netherlands: Kluwer Academic Publishers,

1991.

24. Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions

for applications to soil-structure interaction analysis. Earthquake Spectra 1991; 7(1):1-

28.

25. Lou L, Zerva A. Effects of spatially variable ground motions on the seismic

response of a skewed, multi-span, RC-highway bridge. Soil Dynamics and Earthquake

Engineering 2005; 25: 729-740.

26. Baecher GB, Chan M, Ingra TS, Lee T, Nucci LR. Geotechnical reliability of

offshore gravity platforms. Report MITSG 80-20, Sea Grant College Program,

Cambridge: MIT Press; 1980.


7-1

Chapter 7 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions


Abstract: Previous studies of pounding responses of adjacent bridge structures under

seismic excitation were usually based on the simplified lumped mass model or beam-

column element model. Consequently, only point to point pounding in 1D, usually the axial

direction of the structures, could be considered. In reality, pounding could occur along the

entire surfaces of the adjacent bridge structures. Moreover, spatially varying transverse

ground motions generate torsional responses of bridge decks and these response will cause

eccentric poundings. That is why many pounding damage occurred at corners of the

adjacent decks as observed in many previous earthquakes. A simplified 1D model cannot

capture torsional response and eccentric poundings. To more realistically investigate

pounding between adjacent bridge structures, a two-span simply-supported bridge structure

located at a canyon site is established with a detailed 3D finite element model in the present

study. Spatially varying ground motions in the longitudinal, transverse and vertical

directions at the bridge supports are stochastically simulated as inputs in the analysis. The

pounding responses of the bridge structure under multi-component spatially varying

ground motions are investigated in detail by using the transient dynamic finite element

code LS-DYNA. Numerical results show that the detailed 3D finite element model clearly

captures the eccentric poundings of bridge decks, which may induce local damage around

the corners of bridge decks. It demonstrates the necessity of detailed 3D modelling for

realistic simulation of pounding responses of adjacent bridge decks to earthquake

excitations.

Keywords: pounding response; eccentric pounding; torsional responses; 3D FEM; local

site effect; spatially varying ground motions


7-2

7.1 Introduction


pounding between bridge decks during strong earthquakes is often impossible since the

separation gap of an expansion joint is usually a few centimetres to ensure a smooth traffic

flow. Therefore, pounding damages of adjacent bridge structures have always been

observed in previous major earthquakes. In the 1971 San Fernando earthquake, it was

found that impacts between bridge decks and abutments were the source of extensive

damages to highway bridges with seat type abutments [1]. In the 1989 Loma Prieta

earthquake, poundings between the lower roadway and columns supporting the upper-lever

deck of the Southern viaduct section at the China Basin in California led to significant

damage to the decks and column sides [2]. Reconnaissance reports from the 1995 Kobe

earthquake identified pounding as a major cause of fracture of bearing supports, which

subsequently led to the unseating of bridge decks [3]. Surveys conducted after the 1999

Chi-Chi Taiwan earthquake revealed that 30 bridges suffered some damages due to

poundings at the expansion joints [4]. Poundings between adjacent bridge structures were

also observed in the more recent 2006 Yogyakarta earthquake [5] and 2008 Wenchuan

earthquake [6].

The most straightforward approach to avoid seismic pounding is to provide sufficient

separation distances between adjacent structures. Previous studies on the required

separation distances to avoid seismic pounding between adjacent structures mainly focused

on buildings. Studies on the adjacent bridge structures are relatively less, probably because

with conventional expansion joints it is not possible to provide sufficient separations

between bridge decks while not affecting the smooth traffic flow as mentioned above.

However, with the recent development of modular expansion joint (MEJ) in bridge

engineering, the separation gap can be sufficiently large, which makes avoiding pounding

possible. Hao [7] analysed the effect of various bridge and ground motion parameters on

the relative displacement between adjacent bridge decks, and defined the required seating

length for bridge decks to prevent unseating. Chouw and Hao [8] studied the influence of

soil-structure interaction (SSI) and ground motion spatial variation effects on the required

separation distance of two adjacent bridge frames connected by a MEJ. More recently, Bi et

al. investigated local site effect [9] and SSI [10] on the required separation distances

between bridge structures crossing a canyon site to avoid seismic pounding.


7-3

Pounding is an extremely complex phenomenon involving damage due to plastic

deformation, local cracking or crushing, fracturing due to impact, and friction when two

adjacent bridge decks are in contact with each other. To simplify the analysis, many

researchers modelled a bridge girder as a lumped mass. For example, Malhotra [11]

investigated a concrete bridge that experienced significant pounding during California

earthquakes with a lumped mass model; Jankowski et al. [12] presented an analysis of

pounding between superstructure segments of an isolated elevated bridge induced by the

seismic wave passage effect; Ruangrassamee and Kawashima [13] calculated the relative

displacement spectra of two single-degree-of-freedom (SDOF) systems with pounding

effect; DesRoches and Muthukumar [14] examined the factors affecting the global

response of a multiple-frame bridge due to pounding of adjacent frames; Chouw and Hao

[15, 16] studied the influence of ground motion spatial variation and SSI on the relative

response of two bridge frames. Some other researchers modelled the bridge girders as

beam-column elements. For example, Jankowski et al. [17] discretized the superstructure

segments and piers as 3D elastic beam-column elements, and investigated several

approaches for reducing the negative effects of pounding between superstructure segments

of an isolated elevated bridge. Chouw et al. [18] modelled the girders and piers as 2D beam

elements, and studied the effects of multi-sided poundings on structural responses due to

spatially varying ground motions.

Based on these simplified lumped mass model or beam-column element model, only 1D

point to point pounding, usually in the axial direction of the structures, can be considered.

In a real bridge structure under seismic loading, pounding could take place along the entire

surfaces of the adjacent structures. Moreover, it was observed from previous earthquakes

that most poundings actually occurred at corners of adjacent bridge decks as shown in

Figure 7-1. This is because torsional responses of the adjacent decks induced by spatially

varying transverse ground motions at multiple bridge supports resulted in eccentric

poundings. To more realistically model the pounding phenomenon between adjacent

bridge structures, a detailed 3D finite element analysis is necessary. Zanardo et al. [19]

modelled the box-section bridge girders as shell elements and piers as beam-column

elements, and carried out a parametric study of pounding phenomenon of a multi-span

simply-supported bridge with base isolation devices. Julian et al. [20] evaluated the

effectiveness of cable restrainers to mitigate earthquake damage through connections

between isolated and non-isolated sections of curved steel viaducts using three-dimensional

non-linear finite element response analysis. Although 3D FE models of bridge structures

were developed in those two studies [19, 20], neither the surface to surface nor eccentric


7-4

pounding was considered, instead the pounding was simulated by the contact elements

which linked the external nodes of adjacent segments together. Zhu et al. [21] proposed a

3D contact-friction model to analyse pounding between bridge girders of a three-span steel

bridge. This method overcomes the limitation of the previous studies that pre-define the

pounding locations, therefore provides a more realistic modelling of pounding responses

between bridge decks. The drawback of the method is that it could not model material

non-linearities during contacts. The task to search contact pairs is also very time consuming

and the searching algorithm is complicated. More recently, Jankowski [22] analyzed the

earthquake-induced pounding between the main building and the stairway tower of the

Olive View Hospital based on the non-linear finite element method (FEM), and concluded

that the use of FEM with a detailed representation of the geometry and the non-linear

material behaviour makes the study of earthquake-induced pounding more reliable than

using the discrete lumped mass or beam-column element models. To the best knowledge

of the authors, a simultaneous study of surface to surface, and torsional response induced

eccentric pounding between adjacent bridge structures based on a detailed 3D FEM has

not been reported yet.

Figure 7-1. A typical pounding damage between bridge decks in Chi-Chi earthquake

Pounding between adjacent bridge decks occurs because of large relative displacement

responses. Ground motion spatial variation, besides differences in vibration properties of

adjacent bridge structures, is a source of relative displacement responses. Owing to the

difficulty in modelling ground motion spatial variation, many studies assumed uniform

excitations [11, 13, 14, 20-22] or assumed variation was caused by wave passage effect only

[12, 17]. Only a few studies considered the combined wave passage effect and coherency

loss effect in analyzing relative displacement responses of adjacent bridge structures [15,

16, 18, 19]. It should be noted that all these studies mentioned above assumed that the

analyzed structures locate on a flat-lying site, the influence of local site effect, which further

intensifies ground motion spatial variation at multiple structural supports, are neglected.


7-5

Studies revealed that local site effect not only causes further phase difference [23, 24], but

also affects the coherency loss between spatial ground motions [25]. These differences will

significantly affect the structural responses [9, 10, 24]. Consequently, neglecting local soil

effect on the spatial ground motion variations at multiple supports of a bridge structure

crossing a canyon site may lead to inaccurate estimation of bridge responses.

In this study, pounding responses between the abutment and the adjacent bridge deck and

between two adjacent bridge decks of a two-span simply-supported bridge located on a

canyon site are investigated. A detailed 3D finite element model of the bridge is

constructed in ANSYS [26], and then LS-DYNA [27] is employed to calculate the

structural responses. To model the local site effect on spatial ground motions, the base

rock motions are assumed consisting of out-of-plane and in-plane waves and are modelled

by a filtered Tajimi-Kanai power spectral density function and an empirical coherency loss

function. Seismic waves then propagate vertically through local soil sites to ground surface.

The three-dimensional spatially varying ground motions at different supports of the bridge

structure are then stochastically simulated based on the combined spectral representation

method and the one dimensional wave propagation theory. The simulated spatial ground

motions are used as inputs to calculate structural responses. The influences of pounding

effect, local soil condition and ground motion spatial variation effect on the structural

responses are investigated in detail. It should be noted that the present study concentrates

on modelling the surface to surface pounding and torsional response induced eccentric

pounding. The material non-linearities and pounding induced local damage are not

considered in the present study, which will be included in the subsequent studies.

7.2 Method validation

A multi-span concrete bridge studied by Malhotra [11] is selected to investigate the

reliability of various models used in simulating pounding responses. They are a lumped

mass SDOF model, a beam-column element model by using the contact element to

simulate the pounding effect, and a 3D FE model.

In [11], Malhotra studied the collinear impact between two concrete rods based on the

stereomechanic method, and then applied the procedure to the analysis of pounding effect

of a 300m multi-span concrete bridge separated by an intermediate hinge. The bridge was

simplified as two uncoupled SDOF systems as shown in Figure 7-2(a). The length, mass,

column stiffness and damping ratio for the short span are Ls=100m, ms=1.2×106kg,


7-6

ks=107MN/m and ξs=0.05, respectively. The corresponding parameters for the long span

are Ll=200m, ml=2.4×106kg, kl=94MN/m and ξl =0.05. These parameters correspond to

the vibration frequencies for the short and long span of fs=1.5 and fl=0.996Hz, respectively

[11]. The separation gap between the short and long spans is 5cm.

Using the stereomechanic method, the parameters given above are enough. However, for

the beam-column element model and 3D finite element model, these parameters are

insufficient. Therefore, the following parameters are also used based on the known

properties of the bridge [11]. They are: Young’s modulus of the bridge decks and piers

E=35GPa; densityρ=2400kg/m3; rectangular cross section of the decks m5.22× with

m5.2 in the transverse direction of the bridge; heights of the bridge piers mh 9= , with

cross section m5.2963.0 × and m5.2922.0 × for the short and long span, respectively.

Figure 7-2(b) and (c) shows the beam-column element model and the detailed 3D finite

element model, respectively.

The beam-column model is constructed in ANSYS, and an impact element is used to

model the pounding effect. The stiffness (kp) and damping (cp) of the impact element are

two important parameters that need to be determined. Previous investigation suggested a

kp varying from 10 to 40 times of the lateral stiffness of the stiffer adjacent structures [28].

kp is assumed to be mMN /5000 in the present study as suggested in [16]. The dashpot

constant cp determines the energy dissipated during impact. It is determined by relating it

to the coefficient of restitution (e) at pounding as follows [12]:

ls

lsppp mm

mmkc+

= ζ2 (7-1)

with

( )22 ln

ln

e

ep

+

−=

πζ (7-2)

In the present study, e=0.46 is used [11], which corresponds to a damping ratio of

24.0=pζ .

The 3D finite element model is constructed in ANSYS, but the calculations are carried out

by using LS-DYNA. Eight-node solid elements of size 0.1m are used for both decks and

piers in the model. The treatment of sliding and impact along contact surfaces is an


7-7

important issue in the modelling. To realistically consider the poundings between entire

surfaces of adjacent bridge decks, the contact type CONTACT AUTOMATIC SURFACE

TO SURFACE in LS-DYNA is employed. This contact algorithm is used to avoid

penetration at the contact interfaces.

(a) (b)

(c)

Figure 7-2. Different models (not to scale): (a) lumped mass model (from [11]);

(b) beam-column element model; and (c) 3D finite element model

The bridge is excited in the longitudinal direction only by the first 6.3s of the 1940 North-

South El Centro earthquake ground motion scaled to a peak ground acceleration (PGA) of

0.5g. All materials are assumed as linear elastic in the simulations. Figure 7-3 shows the

structural responses obtained from the different models. As shown in Figure 7-3(a), the

relative displacements in the longitudinal direction between the adjacent bridge decks

obtained by using the lumped mass model [11] are generally smaller than those based on

the beam-column model and detailed 3D finite element model. These results are actually

expected, since the lumped mass model only considers the fundamental vibration mode of

each uncoupled system, the contribution of higher vibration modes are not involved. For

the long-span bridge structure, the vibration frequencies for different vibration modes are

close to each other, the contribution of higher vibration modes could be significant. Both

the beam-column model and 3D model capture the influence of higher vibration modes.

As a result, more high frequency oscillations can be observed in Figure 7-3(a). It also can

be seen that the relative displacements based on these two models are very similar. The

contact forces were not presented in Reference [11] due to the limitation inherent in the


7-8

method, but it was found that poundings occurred at 2.0, 2.7, 3.7, 4.6, 5.5, 6.3 and 7.3s.

Figure 7-3(b) shows the pounding forces based on the beam-column model and the 3D

model. It can be seen that poundings occur at the time instants observed in [11]. The

pounding forces obtained from the 3D model are usually larger than those from the beam-

column model. It should be noted that the pounding force obtained from the beam-

column model depends on the pounding stiffness kp of the impact element, while the

selection of kp is difficult since it depends on many factors and consequently the value can

be varied in a wide range [28]. With a proper selection of kp, closer results are expected.

(a) (b)

Figure 7-3. Structural responses based on different models: (a) relative displacement and

(b) pounding force

Based on the above analysis, it can be concluded that if earthquake ground excitation

occurs only in the longitudinal direction of the bridge, all these three models can be used to

calculate bridge pounding responses. However, the lumped mass model might

underestimate the relative displacements between adjacent bridge decks. The beam-column

model based on the contact element method can give reliable predictions of pounding

responses if a proper pounding element with suitable stiffness and damping ratio is used.

Therefore, if considering only uniaxial ground excitation in the longitudinal direction of the

bridge, detailed 3D model is not necessary as it requires considerably more computational

effort. In reality, however, earthquake ground motion is not limited to only one direction.

Bridge structures inevitably subject to the excitations of multi-component and spatially

varying ground motions. Spatially varying transverse ground motions induce coupled

transverse and torsional responses of bridge decks even the bridge structures are

symmetric. The torsional response might induce eccentric poundings between adjacent

bridge decks as observed in Figure 1, and eccentric poundings in turn will cause more


7-9

torsional responses. This 3D response characteristic cannot be captured with the lumped

mass model or the 2D beam-column model. To realistically model 3D bridge responses

involving possible surface-to-surface and eccentric poundings, the use of a 3D finite

element model is therefore necessary.

7.3 Bridge model

Figure 7-4(a) shows the elevation view of a two-span simply-supported bridge crossing a

canyon site considered in this study. The box-section bridge girders with the cross section

shown in Figure 7-4(b) have the same length of 50m. The Young’s modulus and density of

the girders are 3.45×1010 Pa and 2500 kg/m3, respectively. The L-type abutment is 8.1m

long in the transverse direction and its cross section is shown in Figure 7-4(c). The height

of the rectangular central pier is 20m, with the cross section shown in Figure 7-4(d). The

materials for the two abutments and the pier are the same, with Young’s modulus and

density of 3.0×1010 Pa and 2400 kg/m3, respectively. The two bridge girders are supported

by 8 high-damping rubber bearings. The cross-sectional area and height of rubber layers in

a single bearing are 0.7921m2 and 0.082m. The horizontal effective stiffness and equivalent

damping ratio of a bearing are 2.33×107 N/m and 0.14 respectively [12, 19]. The stiffness

of the bearing in the vertical direction is much larger than those in the horizontal

directions, and is assumed to be 1.87×1010 N/m [19]. To allow for contraction and

expansion of the bridge decks from creep, shrinkage, temperature fluctuations and traffic

without generating constraint forces in the structure, a 5cm gap is introduced between the

abutments and the bridge girders and between the adjacent bridge decks. It is noted that

the lateral side stoppers, which are usually installed in practice, are not considered in the

model. The bridge girders can vibrate freely in the transverse direction (z direction) when

pounding is not involved.

The bridge locates on a canyon site, consisting of horizontally extended soil layers on a

half-space (base rock). The foundations of the bridge are assumed rigidly fixed to the

ground surface and SSI is not involved. Points A, B and C are the three bridge support

locations on the ground surface, the corresponding points on the base rock are A’, B’ and

C’.

The 3D finite element model of the bridge is constructed by using the finite element code

ANSYS [26]. The bridge girders, abutments and pier are modelled by eight-node solid

elements. The bearings are modelled by the spring-dashpot elements. The detailed


7-10

geometric characteristics in Figure 7-4 and the material properties are implemented in the

model. To reduce the required computer memory and computational time, detailed

modelling with fine mesh is only applied to the areas near the contact surfaces. In

particular, detailed modelling with the mesh size of 0.2m is only applied to a length of 1m

from each end of the bridge deck and to a length of 0.6m of the abutments. Beyond this

region, the mesh size in the longitudinal direction is 1m. Figure 7-5 shows fine meshed

areas of the model (the numbers in the circles are the nodes examined in the present study,

which will be discussed in Section 7.5). For a convergence test, a smaller mesh size of 0.1m

around the contact areas is also conducted. Numerical results show that the structural

responses are almost the same for the two different mesh sizes. It should be noted that,

only the linear elastic responses are considered in the present study, smaller mesh size

might be needed if local damages are involved. Figure 7-6 shows the first four vibration

frequencies and the corresponding vibration modes of the bridge. As shown, the first four

vibration frequencies of the bridge equal to 1.081, 1.138, 1.254 and 1.313 Hz for the in-

phase longitudinal (x direction), in-phase transverse (z direction), out-of-phase transverse

and out-of-phase longitudinal vibrations, respectively.

Figure 7-4. (a) Elevation view of the bridge, (b) Cross-section of the bridge girder,

(c) Cross-section of the abutment and (d) Cross-section of the pier (unit: mm)


7-11

Rayleigh damping is assumed in the model to simulate energy dissipation during structural

vibrations.The first two vibration modes is chosen to determine the mass and stiffness

coefficients, because the horizontal displacement in the longitudinal and transverse

directions is of special interest due to its significant importance in the pounding responses.

By assuming the structural damping ratio of 5%, for these two modes, the mass matrix

multiplier is obtained as 0.3483 and the stiffness matrix multiplier is 0.0072. The contact

algorithm of CONTACT AUTOMATIC SURFACE TO SURFACE in LS-DYNA is

employed to model impact between the adjacent structures. The Coulomb friction

coefficient of 0.5 is assumed in the analysis [22].

Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings

(a) f1=1.081Hz (b) f2=1.138Hz

(c) f3=1.254Hz (d) f4=1.313Hz

Figure 7-6. First four vibration frequencies and mode shapes of the bridge

7.4 Spatially varying ground motions

For the canyon site as shown in Figure 7-4, local site will significantly change the

amplitudes and frequency contents of the incoming waves on the base rock owing to the

amplification and filtering effect. The three sites (A, B and C) as shown in the figure have

different influences on base rock motions, thus further intensifies the spatial variations of

1 2

3 4 Left abutment

Left girder 12

10

11

9

7

5 6

8 Left girder Right girder Right girder

Pier

Right abutment


7-12

the ground motions. However, traditional method (e.g., Hao et al. [29]) to simulate the

spatially varying ground motions is based on the flat-lying site assumption and the

influence of local site effect is not considered. With such an assumption, ground motions at

the three sites on ground surface have the same intensity and frequency contents. More

recently, Bi and Hao [25] developed an approach to stochastically simulate the spatially

varying motions on the ground surface of a canyon site. In the method, the base rock

motions are assumed to consist of out-of-plane SH wave and in-plane combined P and SV

waves propagating into the site with an assumed incident angle. The power spectral density

function on the base rock is assumed to be the same, and is modelled by a filtered Tajimi-

Kanai power spectral density function [30]. The spatial variation of ground motions at base

rock is modelled by an empirical coherency function. Local site effect is modelled using the

one-dimensional wave propagation theory [31]. The power spectral density functions of the

horizontal in-plane, horizontal out-of-plane and vertical in-plane motions on the ground

surface can thus be formulated by considering local site effect in the corresponding

directions. The multi-component spatially varying ground motions can then be simulated

by using the approach similar to the traditional method. This approach directly relates site

amplification effect with local soil conditions, and can capture the multiple vibration modes

of local site, is believed more realistically simulating the multi-component spatially varying

motions on surface of a canyon site.

The ground motion intensities at points A’, B’ and C’ on the base rock are assumed to be

the dame and have the following form:

Γ+−

++−

= 222222

222

2222

4

4)(41

)2()()(

ωωξωωωωξ

ωξωωωωω

ggg

gg

fffgS (7-3)

where ωg and ξg are the central frequency and damping ratio of the Tajimi-Kanai power

spectral density function, ωf and ξf are the corresponding central frequency and damping

ratio of the high pass filter function. Γ is a scaling factor depending on the ground motion

intensity. In the analysis, the out-of-plane horizontal motion is assumed to consist of SH

wave only, while the in-plane horizontal and vertical motions are assumed to be combined

P and SV waves. The parameters for the horizontal motion are assumed as πω 10=g rad/s,

6.0=gξ , πω 5.0=f , 6.0=fξ and 0232.0=Γ m2/s3. These parameters correspond to a

ground motion time history with duration T=16s and PGA of 0.5g based on the standard

random vibration method [32]. The vertical motion on the base rock is also modelled with


7-13

the same filtered Tajimi-Kanai power spectral density function, but the amplitude is

assumed to be 2/3 of the horizontal component.


motions at points j’ and k’ (where j’, k’ represents A’, B’ or C’) on the base rock:


appkjappkjappkjkjkj vdivdvdiii αωβωαωωγωγ −⋅−=−= (7-4)

where β is a coefficient reflecting the level of coherency loss. β =0.0, 0.001 and 0.002 are

considered in the present paper, which represent perfectly correlated spatial ground

motions, or spatial ground motion with wave passage effect only, intermediately and weakly

correlated motions, respectively. dj’k’ is the distance between the points j’ and k’. For the

analysed bridge structure, dA’B’=dB’C’=50m, and dA’C’=100m. vapp is the apparent wave

velocity on the base rock, which is related to the base rock property and incident angle α .

With the given properties of local site (shown in Table 7-1) and assumed incident angle o60=α , vapp equals 1697m/s in the present study.

Not to further complicate the problem, only one single layer resting on the base rock is

considered, and the soil properties at sites A, B and C are assumed to be the same, the only

difference is the soil depth. In the present study, the depths for the three local sites are

48.6, 30 and 48.6m respectively. To study the influence of local soil conditions, two types

of soil, i.e. firm and soft soils, are considered. Table 7-1 gives the corresponding parameters

for the soils and base rock. It should be noted that to limit the considered influence factors,

SSI is not considered even when the bridge model locates on a soft soil site.


Type Density (kg/m3) Shear modulus(MPa) Damping ratio Poisson’s ratio

Base rock 2500 1800 0.05 0.33

Firm soil 2000 320 0.05 0.4

Soft soil 1600 40 0.05 0.4

With the proposed approach in [25] and the given parameters of local site, the horizontal

in-plane, horizontal out-of-plane and vertical in-plane motions on the ground surface can

be simulated. It should be noted that a series of random phase angles uniformly distributed

over the range of [0, 2π] are included in the simulation. For each realization of the phase


7-14

angles, one set of ground motion time histories can be simulated. Since most design codes

require 2 to 4 independent analyses with independently simulated ground motions as input

and take the averaged structural responses, in this study, three sets of multi-component

spatially varying ground motions are independently simulated and used as input in the

analysis. In the simulation, the sampling frequency and the upper cut-off frequency are set

to be 100 and 25 Hz respectively, and the time duration is assumed to be T=16s. Figures 7-

7 and 7-8 show the simulated three-dimensional spatially varying acceleration and

displacement time histories on ground surface corresponding to the soft soil conditions

with intermediate coherency loss. Figure 7-9 shows the comparisons of the simulated

power spectral densities with the theoretical values of the horizontal in-plane motions,

good agreements are observed. For conciseness, the comparisons of the horizontal out-of-

plane and vertical in-plane motions are not plotted. Good agreements for these two ground

motion components are also observed. For the coherency loss function between the

motions on the ground surface, Reference [25] indicates that it is different from that on the

base rock. The spatial ground motions on ground surface are least correlated when the

spectral ratios of two local sites differ from each other significantly. Discussion of the

influence of local soil condition on spatial ground motion coherency loss is out of the

scope of the present study. More detailed information can be found in Reference [25]. It

should be noted that the simulated spatial ground motions corresponding to the firm soil

condition also match the model values very well.

Figure 7-7. Simulated acceleration time histories with soft soil condition and intermediately

correlated coherency loss


7-15

Figure 7-8. Simulated displacement time histories with soft soil condition and

intermediately correlated coherency loss

Figure 7-9. Comparison of PSDs between the generated horizontal in-plane motions on

ground surface with the respective theoretical model value


The earthquake-induced pounding responses of the two-span simply-supported bridge as

shown in Figure 7-4 are discussed in detail in this section. The simulated horizontal in-

plane, horizontal out-of-plane and vertical in-plane motions are applied simultaneously

along the longitudinal, transverse and vertical directions of the bridge respectively as shown

in Figure 7-10, where xAd , yAd and zAd represents input displacement time histories in the

x, y and z directions at site A. So as for sites B and C. All the calculations are carried out by

using the transient dynamic finite element code LS-DYNA. The time step is automatically

selected by the code so that converged results can be obtained. To investigate the


7-16

influences of pounding effect, local soil conditions and ground motion spatial variations on

the structural responses, five different cases as shown in Table 7-2 are studied. In which,

the case without pounding (Case 1) is simulated by adjusting the model to make the

separation gaps between the abutment and the girder and between two adjacent girders

large enough so that pounding phenomenon can be completely precluded and the structure

vibrates freely.

dxA

Abutm

ent

dzA

dxC

dzC

dxB

dzB

Girder Girder

Pier

dyB

dyA dyC

Figure 7-10. Multi-components spatially varying inputs at different supports of the bridge

Table 7-2. Different cases studied

Case Soil conditions Coherency loss With/without pounding

1 Firm intermediately without

2 Firm intermediately with

3 Soft intermediately with

4 Firm wave passage effect with

5 Firm weakly with

Poundings may occur between the abutments and the adjacent bridge girders and between

two adjacent bridge girders as mentioned above. Although the bridge considered is a

symmetrical structure, the responses of different parts will be different owing to the ground

motion spatial variations and pounding effects. To obtain a general idea of the earthquake-

induced structural responses, the 12 nodes as indicated in Figure 7-5 are selected to record

the results. Three simulations using the three sets of independently simulated spatially

varying ground motions as inputs for each case are carried out in the present study, the

mean peak responses, which are mostly concerned in engineering practice, are calculated

and discussed. For a better understanding of the results, the time histories of the structural

response corresponding to a particular set of ground motions are also plotted when

necessary.


7-17

7.5.1 Longitudinal response

Figures 7-11, 7-12 and 7-13 show the longitudinal displacement response time histories at

nodes 1 and 2 of different ground motion cases. For conciseness, the response time

histories of other nodes are not plotted. The mean peak displacements at different nodes

are listed in Table 7-3. As shown in Figure 7-11(a), the longitudinal displacement response

of node 1 is almost unaffected by the poundings owing to the fact that the abutment is

quite rigid as compared to the adjacent girder. Similar observations were obtained by

Maragakis et al. [34], who investigated the influences of abutment and deck stiffness, gap,

and deck to abutment mass ratio on the pounding responses between abutments and

bridge decks, and concluded that pounding effect on rigid abutment is not evident. The

influence of collisions on the girder response is, however, significant. As shown in Figure

7-11(b), the peak displacements of node 2 in the longitudinal direction with and without

pounding effect are 0.210 and 0.274m respectively, poundings result in a reduction of

displacement response by 23.4%. This is because the rigid abutment acts as a constraint to

the flexible girder. Comparing the mean peak responses of different nodes of Cases 1 and 2

in Table 7-3, same conclusions can be obtained.

The influence of local soil conditions on the structural response is shown in Figure 7-12.

As shown, softer soil results in lager longitudinal displacement. Taking node 2 for example,

the peak displacements are 0.210 and 0.276m for firm and soft soil respectively. This is

because softer soil usually leads to larger ground displacements at the foundations of the

structure, which results in larger total structural displacement responses. Comparing the

mean peak responses of cases 2 and 3 in Table 7-3, same conclusions can be drawn.

The influence of coherency loss on the longitudinal displacement is shown in Figure 7-13.

As shown in Figure 7-13(a), the influence of coherency loss on node 1 displacement is

insignificant. This is because the ground motions propagate from left to right in the present

study, the simulated ground motion time histories at site A are the same for the three sets

of ground motions of each considered cases. The influence is expected for nodes at the

girders and right abutment. As shown in Figure 7-13(b), different coherency loss results in

different longitudinal displacements of node 2. By examining the mean peak responses of

cases 2, 4, and 5 in Table 7-3, it is generally true that the higher is the correlation between

spatial ground motions, the larger is the longitudinal mean peak responses.


7-18

Figure 7-11. Influence of pounding effect on the longitudinal displacement response

Figure 7-12. Influence of soil conditions on the longitudinal displacement response

Figure 7-13. Influence of coherency loss on the longitudinal displacement response


7-19

Table 7-3. Mean peak displacements in the longitudinal direction (m)

Case Node

1 2 3 4 5

1 0.147 0.148 0.204 0.149 0.151

2 0.216 0.168 0.220 0.177 0.163

3 0.147 0.148 0.203 0.148 0.147

4 0.227 0.176 0.233 0.190 0.173

5 0.200 0.167 0.235 0.182 0.161

6 0.203 0.169 0.234 0.183 0.157

7 0.215 0.173 0.233 0.191 0.158

8 0.226 0.176 0.244 0.177 0.165

9 0.206 0.168 0.231 0.177 0.151

10 0.140 0.140 0.191 0.146 0.138

11 0.205 0.170 0.223 0.169 0.168

12 0.140 0.139 0.191 0.146 0.138

7.5.2 Transverse and vertical responses

As will be demonstrated, the influences of different site and ground motion parameters on

the transverse and vertical displacement responses of the bridge follow the same pattern, so

they are discussed together in this section. Figures 7-14, 7-15 and 7-16 show the response

time histories in the transverse direction of nodes 1 and 2, and the corresponding time

histories in the vertical direction are plotted in Figures 7-17, 7-18 and 7-19. The mean peak

responses in the transverse and vertical directions are listed in Table 7-4 and Table 7-5,

respectively. Similar to the responses in the longitudinal direction, the influence of

poundings on displacement response of the abutments can be neglected. However, the

influence on responses of the bridge girder is evident. Poundings usually result in smaller

peak transverse and vertical displacements. This is because of the friction forces between

the adjacent surfaces during poundings, which reduce the displacement responses of the

bridge structures in the transverse and vertical directions. As shown in Figures 7-15 and 7-

16, Tables 7-4 and 7-5 or the responses in the transverse and vertical directions, softer soil

condition always results in larger displacement responses as discussed above. Ground

motion spatial variations affect bridge responses, especially the responses of bridge decks.

As shown in Tables 7-4 and 7-5, weakly correlated ground motions, among the three

spatial ground motion cases, usually lead to the largest mean peak responses in the two

directions. It also can be seen from Figures 7-17, 7-18 and 7-19 that more high frequency

contents are involved in responses in the vertical direction as compared to those in the

longitudinal and transverse directions. This is because the stiffness of the bridge in the

vertical direction is much higher than that in the longitudinal and transverse directions. For


7-20

the considered bridge model, the first vertical vibration mode is the 7th mode and the

vibration frequency is 2.237 Hz. It should be noted that the lateral side stoppers are not

considered in the present study. If the stoppers are considered, the transverse responses

might be altered.

Figure 7-14. Influence of pounding effect on the transverse displacement response

Figure 7-15. Influence of soil conditions on the transverse displacement response

Figure 7-16. Influence of coherency loss on the transverse displacement response


7-21

Table 7-4. Mean peak displacements in the transverse direction (m)

Case Node

1 2 3 4 5

1 0.119 0.119 0.202 0.119 0.119

2 0.186 0.183 0.265 0.191 0.188

3 0.119 0.119 0.202 0.119 0.119

4 0.185 0.182 0.265 0.189 0.187

5 0.270 0.252 0.349 0.259 0.274

6 0.272 0.230 0.342 0.264 0.271

7 0.270 0.252 0.348 0.259 0.272

8 0.272 0.230 0.341 0.263 0.271

9 0.192 0.189 0.289 0.192 0.201

10 0.123 0.123 0.196 0.119 0.125

11 0.191 0.189 0.289 0.190 0.199

12 0.123 0.123 0.196 0.119 0.125

Figure 7-17. Influence of pounding effect on the vertical displacement response

Figure 7-18. Influence of soil conditions on the vertical displacement response


7-22

Figure 7-19. Influence of coherency loss on the vertical displacement response

Table 7-5. Mean peak displacements in the vertical direction (m)

Case Node

1 2 3 4 5

1 0.072 0.072 0.079 0.072 0.072

2 0.162 0.111 0.131 0.116 0.114

3 0.072 0.072 0.080 0.072 0.072

4 0.145 0.117 0.136 0.118 0.119

5 0.152 0.106 0.127 0.118 0.117

6 0.127 0.115 0.129 0.115 0.121

7 0.131 0.112 0.142 0.117 0.121

8 0.131 0.121 0.130 0.116 0.132

9 0.132 0.109 0.118 0.108 0.122

10 0.073 0.073 0.081 0.073 0.073

11 0.125 0.110 0.121 0.105 0.119

12 0.073 0.073 0.081 0.073 0.072

7.5.3 Torsional response

With the lumped mass model or beam-column element model, the torsional response of

the structure cannot be considered because they are 2D models. With the detailed 3D finite

element model, the torsional responses can be readily estimated. In this study, the torsional

responses are estimated by the rotational angle of the corresponding nodes on both sides

of the same section, i.e., between nodes 1 and 3, nodes 2 and 4, etc. These can be achieved

by dividing the relative longitudinal displacement of these corresponding nodes by the deck

width, which is 8.1m in the present study. Table 7-6 shows the mean peak rotational angles

for different cases. Different from the longitudinal, transverse and vertical displacement

responses, poundings increase the torsional responses. This is because pounding imposes a


7-23

restraint to the bridge spans, thus reduces lateral responses. However, eccentric poundings

induced by spatially varying ground motions generate large eccentric impact forces that

enhance the torsional responses. Comparing Case 3 with Case 2, it is obvious again that

softer soil results in larger torsional responses. Comparing the responses obtained from

spatial ground motions with different coherency losses, it is difficult to draw a general

conclusion. Although highly correlated ground motions usually lead to the largest

longitudinal displacements as discussed in Section 7.5.1, they do not necessarily yield the

largest torsional response. This is probably because the torsional response is related to the

relative displacement between nodes on the same cross section of the bridge structure

instead of the absolute displacement.

To examine the occurrence of poundings, the longitudinal displacements of nodes 1 and 2

and nodes 3 and 4 are plotted in the same figure with the displacements of nodes 1 and 3

shifted by the initial gap of 5cm. Thus, in the figure, the instants when the displacements of

the two adjacent points coinciding with each other indicate the occurrence of poundings.

As shown in Figure 7-20(a), node 1 and node 2 come into contacts 15 times, at the time

instants 3.26, 5.29, 6.29, 6.68, 7.30, 7.72, 8.20, 8.63, 9.13, 9.66, 11.13, 11.89, 12.44, 13.70

and 14.26s. Whereas between nodes 3 and 4 as shown in Figure 7-20(b), the poundings at

6.29, 11.89 and 12.44s do not occur, but two more collisions can be observed at 3.76 and

13.20s. Since these points locate at the opposite corners of the bridge deck cross section,

pounding at these points occurring simultaneously implies the entire cross sections are in

contact, i.e. surface to surface pounding occurs. Otherwise, they are torsional response

induced eccentric poundings. In this example, pounding occurring at 6.29, 11.89 and 12.44s

are eccentric poundings between nodes 1 and 2, and those at 3.76 and 13.20s are eccentric

poundings between nodes 3 and 4. Torsional response induced eccentric poundings

between other corner points shown in Figure 7-5 are also observed. Owing to page limit,

they are not shown here. These observations indicate that if 3D model with tri-axial ground

motion inputs are considered, more number of poundings will be observed than the

lumped mass and 2D beam-column element model because the two letter models cannot

capture the possible eccentric poundings induced by torsional responses.


7-24

Table 7-6. Mean peak rotational angle (degree)

Case Node

1 2 3 4 5

1 and 3 0.0014 0.0177 0.0219 0.0262 0.0149

2 and 4 0.2638 0.2957 0.3459 0.2745 0.3027

5 and 7 0.2150 0.2504 0.3374 0.2879 0.2844

6 and 8 0.2271 0.2624 0.3317 0.2214 0.2766

9 and 11 0.2624 0.3211 0.3572 0.2822 0.2872

10 and 12 0.0007 0.0170 0.0198 0.0113 0.0127

Figure 7-20. Longitudinal displacements of different nodes to case 2 ground motion

7.5.4 Resultant pounding force

Resultant pounding force in the longitudinal direction can be obtained by integrating the

normal stresses over the entire cross section of the contact surface. Though torsional

response induced eccentric poundings may result in the noncollinear impacts on the

contact surface, the components of pounding forces in the transverse and vertical

directions, which are induced owing to frictional forces during contact, are relatively small

as compared to the component in the longitudinal direction. In this paper, only the

influences of site conditions and coherency losses on the resultant pounding forces in the

longitudinal direction are discussed. Figures 7-21 and 7-22 show the pounding forces at

different time instants corresponding to different ground motion cases. It can be seen from

Figure 7-21 that soft soil condition results in larger peak pounding forces than firm soil

condition. This is because soft soil leads to larger displacement response in the longitudinal

direction as shown in Figure 7-12, which also results in larger relative displacement

between the adjacent components of the bridge and makes the poundings more severe

than that on the firm site. Comparing Figure 7-21(a) and 7-21(c) with 7-21(b), it is obvious

Initial gap=5cm

Initial gap=5cm


7-25

that the pounding forces between two bridge girders are generally smaller than those

between the left or right abutment and the adjacent girder. This is because the bridge

analysed in the present study is a symmetric structure, the left and right girders have the

same dynamic characteristics and tend to vibrate in phase. If the spatially varying ground

motions and the restraints from the abutments are not considered, the two spans will

vibrate fully in phase and no pounding will be observed [7]. At the left and right gaps

between abutment and girder, the abutments are much rigid than the adjacent bridge

girders, the relative displacement is induced not only by spatially varying ground motions,

but also by out of phase vibrations owing to different vibration frequencies of abutment

and bridge span. In this case, the out of phase vibration induced relative displacement

response dominates the responses. Therefore, larger pounding forces between abutments

and girders are observed. Figure 7-22 illustrates the consequence of coherency loss between

spatial ground motions for the pounding force development. As shown, spatially varying

ground motions with wave passage effect only lead to larger pounding forces. This also can

be explained by its influence on the longitudinal displacements as shown in Figure 7-13 and

Table 7-3, where wave passage effect results in larger relative displacement responses. Same

conclusion was also drawn in [16], in which the two adjacent bridge girders were simplified

as two lumped masses.

Figure 7-21. Influence of soil conditions on the resultant pounding forces


7-26

Figure 7-22. Influence of coherency loss on the resultant pounding forces

7.5.5 Stress distributions

By using the traditional lumped-mass model or beam-column element model, the stress on

the entire contact surface will be the same. However, the use of 3D finite element model

allows a more detailed prediction of the largest stresses and their locations, and thus where

earthquake-induced damage may occur. Figure 7-23 shows the stress distributions in the

longitudinal direction at left expansion joint of the bridge corresponding to the different

cases considered in this study at the time instant when peak resultant pounding force

occurs. As shown in Figure 7-23(a), when bridge is on the firm soil site, the maximum

compressive stress appears at the bottom outside corner of the girder. However, when it is

on the soft soil site, the maximum compressive stress appears at the top inside corner of

the girder. Although surface to surface pounding occurs, the largest stresses always occurs

at the corners of the bridge girders corresponding to eccentric poundings because the

pounding forces are distributed in a smaller area. This is why most observed pounding

damages occurred at corners of bridge girders. It also can be seen that larger resultant

pounding force not necessarily results in larger compressive stress. Taking the results from

different soil conditions as example, the peak resultant pounding force for firm and soft

soil are 55 and 80 MN, respectively as shown in Figure 7-21. The resultant pounding force

corresponding to the soft site condition is much larger than that corresponding to the firm

site condition. However, the maximum compressive stresses are 88.8 and 59.3 MPa,


7-27

respectively for these two particular pounding events. This is again because the stress

development is not only related to the pounding force but also related to the actual contact

area at each pounding instant. The lumped mass and beam-column element models, which

estimate the stress by dividing the pounding forces by the cross sectional area of the bridge

girder, may not lead to correct predictions of stresses. As also shown in Figure 7-23, the

maximum stresses can reach as high as 105.4 MPa (Figure 7-23(d)). It is much larger than

the compressive strength of normal concrete used in bridge construction, which is usually

30-65 MPa under impact loading [35], thus concrete damages are expected although the

concrete compressive strength increases owing to strain rate effect. These results are

consistent with the observations in the past major earthquakes, in which the damages

around the corners of the structure were usually the most serious as shown in Figure 7-1.

However, it should be noted that only linear elastic responses are considered in this study.

Further study by modelling concrete damage is necessary as concrete damage will affect the

subsequent bridge responses.

(a) (b)

(c) (d)

Figure 7-23. Stress distributions in the longitudinal direction at left gap of different cases at

the time when peak resultant pounding force occur (a) Case 1 at t=6.27s, (b) Case 3 at

t=7.63s, (c) Case 4 at t=7.96s and (d) Case 5 at t=8.04s (unit: Pa)


7-28

7.6 Conclusions

Based on a detailed 3D finite element model, the earthquake-induced pounding responses

between adjacent components of a two-span simply-supported bridge structure located at a

canyon site are studied in the present paper. The influences of local soil conditions and

ground motion spatial variations on the pounding responses are investigated in detail.

Following conclusions are obtained based on the numerical results:

1. The lumped mass model and beam-column element model can be used to calculate

bridge pounding responses if only longitudinal ground excitation is considered. The

detailed 3D finite element model is necessary to model the torsional response

induced by spatially varying transverse ground motions and the corresponding

eccentric poundings.

2. The influence of pounding effect on the displacement response of the stiff

abutments can be neglected. Its influence on the bridge girder displacement is

evident. Poundings usually result in smaller mean peak displacements in the

longitudinal, transverse and vertical directions, but larger mean peak torsional

responses.

3. Local soil conditions significantly influence the structural responses. The softer is

the local site, the larger are the structural responses.

4. Spatially varying ground motions with wave passage effect only usually lead to

larger longitudinal displacement. Weakly correlated ground motions result in larger

transverse and vertical responses.

5. Maximum stress usually appears at the corners of the contact surfaces owing to

eccentric poundings.

6. 3D FE model is needed for more realistic predictions of pounding responses and

pounding induced bridge girder damages.

7.7 References

1. Jennings PC. Engineering features of the San Fernando Earthquake of February 9,

1971. Report No. EERL-71-02, California Institute of Technology, 1971.

2. Priestley MJN, Seible F, Calvi GM. Seismic design and retrofit of bridges. Wiley: New

York, 1996.



Engineering JSCE 1996; 13(2):211-240.


7-29


Reconnaissance Report. Report No. 01-02, EERI, Oakland, CA, 1999.


2006. Mid-America Earthquake Centre. Report No. 07-02, 2007.


earthquake on bridges. The 14th world conference on earthquake engineering, Beijing, China,

2008; S31-006.


earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1):91-103.



Structures 2008; 30(1):154-162.




10. Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site


seismic pounding. Earthquake Engineering and Structural Dynamics 2010 (published

online).

11. Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete

bridges. Journal of Engineering Mechanics 1998; 124(7):794-802.



1998; 27:487-502.



1538.


response of multi-frame bridges. Journal of Structural Engineering (ASCE) 2002; 128(7):

860-869.



23:717-728.





7-30


during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29: 195-212.

18. Chouw N, Hao H, Su H. Multi-sided pounding response of bridge structures with

non-linear bearings to spatially varying ground excitation. Advances in Structural

Engineering 2006; 9(1):55-66.




20. Julian FDR, Hayashikawa T, Obata T. Seismic performance of isolated curved steel

viaducts equipped with deck unseating prevention cable restrainers. Journal of

Constructional Steel Research 2006; 63:237-253.

21. Zhu P, Abe M, Fujino Y. Modelling three-dimensional non-linear seismic


Engineering and Structural Dynamics 2002; 31:1891-1913.

22. Jankowski R. Non-linear FEM analysis of earthquake-induced pounding between

the main building and the stairway tower of the Olive View Hospital. Engineering

Structures 2009; 31:1851-1864.



24. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially


25. Bi K, Hao H. Influence of irregular topography and random soil properties on


Structural Dynamics 2010 (published online).

26. ANSYS. ANSYS user’s manual revision 12.1. ANSYS Inc, USA, 2009.

27. LS-DYNA. LS-DYNA user manual. Livermore Software Technology Corporation:

California, USA, 2007.

28. Hao H, Ma G. An investigation of the coupled torsional-pounding responses of

adjacnet asymmetric structures. Proceeding of the 7th East Asian-Pacafic Conference on the

Structural Engineering and Constructuion, Kochi, Japan, 1999; 788-793.



111(3): 293-310.





7-31

31. Wolf JP. Dynamic soil-structure interaction. Prentice Hall: Englewood Cliffs, NJ, 1985.

32. Der Kiureghian A. Structural response to stationary excitation. Journal of the

Engineering Mechanics Division 1980; 106(6): 1195-1213.

33. Sobczky K. Stochastic wave propagation. Netherlands: Kluwer Academic Publishers,

1991.

34. Maragakis E, Douglas B, Vrontinos S. Classical formulations of the impact between

bridge deck and abutments during strong earthquake. Proceedings of the 6th Canadian

Conference on Earthquake Engineering, Toronto, Canada, 1991; 205-212.

35. Bischoff PH, Perry SH. Impact behaviour of plain concrete loaded in uniaxial

compression. Journal of Engineering Mechanics 1995; 121(6): 685-693.


8-1

Chapter 8 Concluding Remarks

8.1 Main findings

This thesis has focused on the modelling of spatial variation of seismic ground motions,

and its effect on bridge structural responses. This effort brings together various aspects

regarding the modelling of seismic ground motion spatial variations caused by incoherence

effect, wave passage effect and local site effect, bridge structure modelling with SSI effect,

and dynamic response modelling of bridge structures with pounding effect. The major

contributions and findings made in this research are summarised below.

1. A stochastic method is adopted and further developed in Chapter 2 to investigate the

combined ground motion spatial variation effect and local site effect on the responses

of a bridge frame located on a canyon site. In the proposed approach, the spatial ground

motions are modelled in two steps. Firstly, the base rock motions are assumed to have

the same intensity and are modelled with a filtered Tajimi-Kanai power spectral density

function and an empirical spatial ground motion coherency loss function. Then, power

spectral density function of ground motion on surface of the canyon site is derived by

considering the site amplification effect based on the one dimensional seismic wave

propagation theory. The structural responses are formulated in the frequency domain,

and the mean peak responses are estimated based on the standard random vibration

method. Numerical results show that wave propagation through multiple sites with

different site conditions cause further variations of spatial ground motions, and thus

significantly influence the structural responses.

2. Chapters 3 and 4 investigate the minimum total gaps between abutment and bridge deck

and between two adjacent bridge decks connected by MEJs to avoid seismic pounding

during strong earthquakes. In particular, Chapter 3 focuses on the combined ground

motion spatial variation and local site effect and Chapter 4 highlights the SSI effect. The

stochastic structural responses are also formulated in the frequency domain. Numerical


8-2

results show that the required MEJ total gap depends on the dynamic properties of the

participating adjacent structure and the dynamic behaviour of the supporting subsoil and

the spatially varying ground excitations. Sufficient total gap of a MEJ needed in the

bridge design to preclude possible pounding during strong earthquakes is presented. The

numerical results obtained in these studies can be used as references in designing the

total gap of MEJs.

3. Chapter 5 presents a method to model and simulate spatially varying earthquake ground

motion time histories at sites with non-uniform conditions. It takes into consideration

the local site effect on ground motion amplification and spatial variations. The base

rock motions are modelled by a filtered Tajimi-Kanai power spectral density function or

a stochastic ground motion attenuation model. The specific site ground motion power

spectral density function is derived by considering seismic wave propagations through

the local site by assuming the base rock motions consisting of out-of-plane SH wave

and in-plane combined P and SV waves with an incident angle to the site. The spectral

representation method is used to simulate the spatially varying earthquake ground

motions. It is proven that the simulated spatial ground motion time histories are

compatible with the respective target power spectral densities or design response

spectra individually, and the model coherency loss function between any two of them.

This method directly relates site amplification effect with local soil conditions, and can

capture the multiple vibration modes of local site, is believed more realistically

simulating the multi-component spatially varying motions on surface of a canyon site.

The simulated time histories can be used as inputs to multiple supports of long-span

structures on non-uniform sites in engineering practice.

4. Chapter 6 evaluates the influence of local site irregular topography and random soil

properties on the coherency function between spatial surface motions based on the

method proposed in Chapter 5. In the analysis, the random soil properties are assumed

to follow normal distributions and are modelled by the one-dimensional random fields

in the vertical directions. For each realization of the random soil properties, spatially

varying ground motion time histories are generated and mean coherency loss functions

are derived. Numerical results show that the coherency function between surface

ground motions on a canyon site is different from that between base rock motions. The

lagged coherency function on the base rock is the upper bound of that on the ground

surface. For a canyon site, the coherency function of spatial surface ground motions

oscillates with frequency. The maximum and minimum coherency values are related to

the spectral ratios of two local sites or two wave paths. The coherency function models

for motions on a flat-lying site cannot be used to model that of motions on a canyon


8-3

site. The influence of random soil properties on the lagged coherency function depends

on the level of variations of soil properties.

5. Chapter 7 investigates the pounding responses between the abutment and the adjacent

bridge deck and between two adjacent bridge decks of a two-span simply-supported

bridge located on a canyon site based on a detailed 3D finite element model. The multi-

component spatially varying ground motions are stochastically simulated as inputs based

on the method proposed in Chapter 5, and the numerical analysis is carried out by using

the transient dynamic finite element code LS-DYNA. The influences of local soil

conditions and ground motion spatial variations on the pounding responses are

investigated in detail. Numerical results indicate that the torsional response of bridge

structures resulted from the spatially varying transverse motions induces eccentric

poundings between adjacent bridge structures. Traditionally used SDOF model and 2D

finite element model of bridge structures could not capture the torsional response

induced eccentric poundings, therefore might lead to inaccurate pounding response

predictions. Detailed 3D finite element model clearly captures the eccentric poundings

of bridge decks and the potential bridge deck damages, thus is needed for a more

reliable prediction of earthquake-induced pounding responses between adjacent

structures.

8.2 Recommendations for future work

The modelling of seismic ground motion spatial variations and its effect on bridge

structural responses have been carried out in this research. Further investigations can be

made in the future study as outlined below:

1. Three different computer programs have been developed to investigate the structural

responses in the frequency domain in Chapters 2-4. These programs are based on the

simplified structural models and stochastic structural response analysis. They are thus

only suitable for linear elastic response analysis of the particular problems discussed in

the corresponding chapters. Furthermore, only one-dimensional seismic excitation,

which is along the longitudinal direction of the structure is considered because they are

the primary sources for relative displacement responses of bridge structures in the

longitudinal direction.

2. In Chapter 5, the surface motions of a canyon site with multiple soil layers are derived

based on the one-dimensional wave propagation theory, the scattering and diffraction


8-4

effect of waves by canyons are not involved. How to incorporate this 2D wave

propagation phenomenon into the simulation technique needs be studied.

3. For a canyon site, the coherency function of spatial surface ground motions is found to

be related to the spectral ratios of two local sites in Chapter 6, but analytical relation that

can be straightforwardly used to model the influences of local site conditions on spatial

ground motion coherency loss is not derived yet.. Further study is needed to develop

the analytical relation for easy use in engineering application to predict local site effects

on ground motion spatial variations.

4. Random soil properties are modelled by the independent one-dimensional random fields

in the vertical direction in Chapter 6, the soil nonlinearities are not considered. Soil

nonlinearities also affect the surface motion spatial variations. Further study to

investigate the influence of soil nonlinearities on the surface motion spatial variations is

needed.

5. Surface to surface pounding and torsional response induced eccentric pounding

between different components of a two-spam simply-supported bridge are investigated

in Chapter 7. The material non-linearities and pounding induced local damage are not

considered, which needs to be included in the subsequent studies.

6. Shaking table tests on scaled bridge models to spatially varying earthquake ground

motions to verify the numerical results are necessary.

7. Further study to develop design guides to mitigate pounding and unseating damage of

bridge decks is needed.

Documents

Effects of Ground Motion Spatial Variations and Random Site Conditions … · could not capture the torsional response induced eccentric poundings, therefore might lead to inaccurate